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Exchange Rate Misalignment and External Imbalances: What is the Optimal Monetary Policy Response?

Giancarlo Corsetti Cambridge University and CEPR

Luca Dedola European Central Bank and CEPR

Sylvain Leduc Federal Reserve Bank of San Francisco

February 2020

Working Paper 2020-04

https://www.frbsf.org/economic-research/publications/working-papers/2020/04/

Suggested citation: Corsetti, Giancarlo, Luca Dedola, Sylvain Leduc. 2020. “Exchange Rate Misalignment and External Imbalances: What is the Optimal Monetary Policy Response?” Federal Reserve Bank of San Francisco Working Paper 2020-04.

The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.

Exchange Rate Misalignment and External Imbalances: What

is the Optimal Monetary Policy Response?

Giancarlo CorsettiCambridge University and CEPR

Luca DedolaEuropean Central Bank and CEPR

Sylvain LeducFederal Reserve Bank of San Francisco

This version: January 2020∗

Abstract

How should monetary policy respond to capital inflows that appreciate the currency,

widen the current account deficit and cause domestic overheating? Using the workhorse

open-macro monetary model, we derive a quadratic approximation of the utility-based global

loss function in incomplete market economies, solve for the optimal targeting rules under

cooperation and characterize the constrained-optimal allocation. The answer is sharp: the

optimal monetary stance is contractionary if the exchange rate pass-through (ERPT) on

import prices is incomplete, expansionary if ERPT is complete–implying that misalignment

and exchange rate volatility are higher in economies where incomplete pass through contains

the e§ects of exchange rates on price competitiveness.

Keywords: Currency misalignments, trade imbalances, asset markets and risk sharing,

optimal targeting rules, international policy cooperation, exchange rate pass-through

JEL codes: E44, E52, E61, F41, F42

∗We thank for comments, without implicating, our discussants Gianluca Benigno, Rodrigo Caputo, HarrisDellas, Charles Engel, Jordi Galì, Paolo Pesenti, Bruce Preston, Assaf Razin, Alan Sutherland, Cedric Tille, theparticipants at the CEPR ESSIM, the Bank of Canada-ECB Workshop on Exchange Rates, the 2019 Bank ofJapan-IMES annual conference, the International Research Forum on Monetary Policy, the 2018 InternationalMini Conference in Notre Dame, the World Congress of the Econometric Society, the SNB conference on MonetaryPolicy Advances, the Barcelona Summer Forum: International Capital Flows, and seminar participants at theBanco de Chile, Bank of England, Bank of Spain, Berkeley, Bocconi University, European Central Bank, the2017 ECB Cluster 2 Meeting in Madrid, HEC Montreal, International Monetary Fund, National University ofSingapore, Oxford, Rennes and Riksbank. A previous version of this paper was circulated with the title “ExchangeRate Misalignment, Capital Flows, and Optimal Monetary Policy Trade-o§s”. We thank Emile Marin and SimonLloyd for superb research assistance. Giancarlo Corsetti acknowledges the generous support of the DuisenbergFellowship at the European Central Bank, the Keynes Fellowship at Cambridge University, and the Cambridge-INET Institute at Cambridge. The views expressed in this paper are our own, and do not reflect those of theEuropean Central Bank or its Executive Board, the Federal Reserve System, or any institution with which weare a¢liated.

1

1 Introduction

Cross-border capital flows raise widespread concerns about their potential adverse e§ects on do-

mestic economies. Because of their impact on the exchange rate, domestic demand, and current

account imbalances, inflows and outflows of capital may give rise to challenging policy trade-o§s

between internal objectives (inflation and output gap) and external objectives (competitiveness

and trade). The debate on the most appropriate tools for managing capital movements and

their macroeconomic impact has led to a reconsideration of the role of monetary policy not just

as a complement to other policy instruments (ranging from macroprudential policy to capital

controls) but also as a first-line defense in the absence of other readily implementable tools.

How should a central bank react, if at all, to capital inflows that deteriorate the current

account imbalance and appreciate the currency? One leading answer is that the natural rate still

provides a reliable compass for monetary policy: to the extent that an external deficit raises the

natural rate of interest, capital inflows should be systematically matched by a tighter monetary

stance (see, e.g., Obstfeld and Rogo§ 2010).1 However, this answer may not be statisfactory

in the presence of financial market imperfections and nominal rigidities, whereas, as recently

stressed by Farhi and Werning [2016], pecuniary and demand externalities result in capital flows

and deficits that are ine¢cient (implying either over- or underborrowing), and exchange rates

that are misaligned (i.e., either overvalued or undervalued). So, to the extent that monetary

tightening exacerbates overvaluation, a contraction may not be the optimal policy response

to a capital inflow. Are there situations that call for curbing exchange rate variability and

misalignment, even if this comes at the cost of imperfect stabilization of inflation and output

gaps?

In this paper, we provide an answer to these questions by working out an analytically

transparent characterization of the optimal monetary policy under commitment and cooperation

using the workhorse open economy monetary model–the two-country New Keynesian model.

As a standard and tractable way to introduce ine¢cient capital flows, we assume that the

only internationally traded asset is an noncontingent bond (as in the seminal contribution by

Obstfeld and Rogo§ [1995]; see also Costinot et al. [2015]).2 Our key finding is that the optimal

policy response to capital flows vary systematically depending on the equilibrium response

of misalignment and cross-country demand to these flows–in turn a function of a few key

structural features of open economies–and the degree of exchange rate pass-through (ERPT).

Specifically, we show that, the optimal monetary policy stance in response to ine¢cient cap-

ital inflows associated with an overvalued currency and a demand boom depends on ERPT, i.e.,

on whether export prices are sticky in the currency of the producer (producer currency pricing

or PCP) or in the currency of the importer (local currency pricing, or LCP).3 In economies

in which incomplete ERPT (due to LCP) mutes the e§ects of the exchange rate overvaluation

1“Better macro performance comes from a monetary rule that recognizes how an external deficit raises thenatural real rate of interest.” Obstfeld and Rogo§ [2010] p. 34. See also the recent discussion by Obstfeld [2019]stressing a similar point.

2 In the tradition of Obstfeld and Rogo§ [1995], we capture the lack of e¢cient diversification in the datadespite the number of seemingly available cross-border assets, by focusing on bond economies.

3We focus here on the two symmetric cases of ERPT, which have been so far center stage in the literatureon the optimal design of monetary policy in open economies, see Engel [2011]. In ongoing work we analyze thekey asymmetric case of dominant currency pricing (DCP) recently emphasized by Gopinath [2016], which in atwo-country setting requires a separate, systematic analysis also under complete markets.

2

on the output gap, the optimal stance is contractionary. Since with incomplete ERPT the ex-

change rate has limited expenditure switching e§ects on the composition of demand, it reduces

the importance of stabilizing the exchange-rate misalignment relative to stabilizing aggregate

demand and inflation. As the optimal policy focuses on the latter objectives, it exacerbates the

misalignment–causing the real exchange rate to be more volatile than under a policy regime

of strict CPI stability. Conversely, in economies in which ERPT is complete (PCP), the opti-

mal stance is expansionary and leans against the overvaluation of the currency, at the cost of

some overheating. In this case, the optimal policy reduces the volatility of the currency and

the output gap relative to the natural rate allocation associated with a policy of strict price

stability.

The direction of the optimal policy response is instead independent of ERPT in economies

in which, in equilibrium, ine¢cient capital inflows and deficits are associated with an under-

valued currency and a relatively weak domestic demand–a case that can arise when domestic

and foreign goods are complements, i.e., the trade elasticity is su¢ciently low. Vis-à-vis an

undervalued currency and an ine¢ciently low domestic demand, the optimal policy response to

capital inflows is invariably expansionary. In this case, as monetary policy moves in support of

domestic economic activity, it actually exacerbates exchange rate misalignment and volatility

for any degree of ERPT, relative to strict price stability.

In developing our analysis, we make three specific contributions to the literature. First, we

provide a second-order accurate approximation of the global welfare function for the standard

New Keynesian two-country model with generically incomplete markets under PCP and LCP.4

The derivation of this function does not rely on specific forms of market incompleteness (e.g.,

bond economies and financial autarky obtained as special cases), nor on restrictive assumptions

about preferences (e.g., it is not restricted to the case of unitary trade elasticity or to having

the same consumption baskets across countries).

Second, the paper derives optimal targeting rules under cooperation and commitment for

both PCP and LCP economies. These rules hold for a wide range of shocks (including an-

ticipated or unanticipated shocks to preferences, productivity, markups, etc.), but, unlike the

global welfare function, are specific to bond economies. Based on these rules, we provide an

analytical characterization of the macroeconomic dynamic response to ine¢cient flows under

the optimal policy.

In addition to output gaps and inflation rates, both the welfare function and targeting rules

can be written as a function of real exchange rate misalignment and relative demand misallo-

cation, themselves a function of ine¢cient capital flows. Di§erent from the case of complete

markets, where misalignment and demand misallocation are proportional to each other, these

distortions, combined, define a gap specific to imperfect risk sharing, which we dub “wealth

gap.” As a third contribution, we show that this gap is a direct synthetic measure of the distor-

tions associated with ine¢cient flows and plays a key role in optimal policy design. It acts much

like an endogenous “markup” shock–giving rise to meaningful trade-o§s between inflation, out-

4 In our analysis we abstract from the question of which export pricing strategy, PCP or LCP, is optimal fromthe vantage point of the firms, given the optimal policy (see recent work by Mukhin 2018). An important issuefor future research is whether, in economic environments supporting the optimal choice of either PCP or LCP,the optimal stabilization rules would substantially deviate from the one we derive in this paper.

3

put gaps, demand misallocation and misalignment;5 most crucially, the wealth gap characterizes

whether ine¢cient capital inflow leads to a positive (negative) wealth gap and overvaluation

(undervaluation). In particular, we derive thresholds for the trade elasticity at which the equi-

librium link between ine¢cient flows and misalignment switches sign. These thresholds di§er

across PCP economies, where they are a function of openness, and LCP economies, where they

depend both on openness and the degree of nominal rigidities (in both cases, however, the

thresholds are bounded above by 1/2 under home bias in consumption).

To discuss macroeconomic dynamics, we find it analytically convenient to focus on “news

shocks” (anticipation of future changes in fundamentals) as these typically generate capital flows

that are excessive relative to the first best.6 The news shocks may stem from political risk (i.e.,

capital controls; see, e.g., Acharya and Bengui [2016]), changes in the e¢ciency of financial

intermediaries (see, e.g., Gabaix and Maggiori [2015]), changes in technology or preferences

impinging on savings–without loss of generality, we focus on the latter. Notably, we show

that in model specifications often adopted by the literature (see, e.g., Clarida et al. 2002

and Engel 2011), capital flows in response to news shocks are exogenous to monetary policy.

We can thus bring our analysis to bear directly on a case often debated in policy circles,

where monetary policy can only mitigate the e§ects of ine¢cient capital flows on domestic

macroeconomic dynamics, but cannot curb their size. 7

Related literature Our analysis builds on a vast body of work that, over the last two decades,

has reexamined a classic question in open economy macroeconomics, concerning the trade-o§s

between external and internal objective (see Benigno and Benigno [2003]; Clarida, Galí and

Gertler [2002]; Corsetti and Pesenti [2005]; Devereux and Engel [2003]; Engel [2011]; and Galí

and Monacelli [2005], among others).8 It is nonetheless useful to emphasize two strands of this

literature that help highlight our contribution.

The first is the literature epitomized by Engel [2011], who studies optimal policy under

complete markets contrasting LCP and PCP in the otherwise canonical open economy New

Keynesian model developed by Clarida, Galí and Gertler [2002]. A key result under LCP is that

the optimal monetary policy supports an allocation with CPI-price stability and no exchange

rate misalignment–which also implies no cross-country misallocation of demand—the demand

gap defined in Section 3.1 below. Indeed, under the maintained assumption of complete markets,

trade in financial assets ensures that real exchange rate misalignment and the demand gap

are always proportional to each other–independently of whether ERPT is complete (PCP)

5Moreover, while the exogenous markup shocks typically assumed in the monetary literature create aggregateglobal distortions, we show that the ine¢ciencies from capital inflows have opposing e§ects on di§erent economies,that cancel out in the aggregate. A key implication is that, under the optimal policy, the Home and Foreignmonetary stance will be symmetric but with the opposite sign. This is in contrast with the optimal response tothe exogenous markup shocks commonly assumed by the monetary literature, which may be symmetric acrossborders, in particular under LCP, even when uncorrelated across countries (see e.g. Corsetti et al 2010 page902-904).

6See the seminal papers by Beaudry and Portier [2006] and Schmitt-Grohe and Uribe [2012].7These results are not a§ected by intermediation costs associated to the accumulation of net foreign asset

position. Hence, barring additional algebraic complexity, they extend to economic environments such as the onestudied by Gabaix and Maggiori 2016.

8As discussed in Corsetti, Dedola, and Leduc [2010], most of the papers in the literature either assumecomplete markets or close to e¢cient capital flows because of particular restrictions on preference and technologyparameters.

4

or incomplete (LCP).9 This is where our results di§er from, and complement, this literature.

When markets are not complete, misalignment and demand gaps are not proportional to each

other–monetary policy will not be able to close both of them simultaneously, facing trade-o§s

between competing internal and external objectives.

The second strand of the literature includes a small number of contributions that, like ours,

provide analytical characterizations of the optimal monetary policy in two-country models with

incomplete financial markets.10 Obstfeld and Rogo§ [2003] and Devereux [2004] examine static

frameworks without capital flows, and in which prices are set one period in advance–therefore,

necessarily abstracting from the welfare implications of current account dynamics and inflation.

Devereux and Sutherland [2008] study a dynamic setting similar to ours, but in which markets

are e§ectively complete under flexible prices so that price stability also attains the first-best

natural rate allocation.11 Under PCP, Benigno [2009] emphasizes deviations from price stability,

in economies in which net foreign asset holdings are asymmetrical in the nonstochastic steady

state. However, the focus is on economies in which deviations from both purchasing power parity

(PPP) and the law of one price are assumed away, in contrast with the analysis of real exchange

rate misalignment at the core of optimal policy design analyzed in our paper. Our paper is

also closely related to Farhi and Werning [2016], which provides a general characterization of

optimal targeting rules in economies with nominal rigidities and financial market frictions. While

in their contribution these authors focus on the role of macroprudential policies when monetary

policy is constrained, we focus on optimal monetary policy when macroprudential policies are

not available–taking into account standard welfare costs of inflation that stem from staggered

price setting. Monetary policy with incomplete financial markets is also analyzed quantitatively

in recent work by Rabitsch [2012], who revisits the benefits from international cooperation, and

Senay and Sutherland [2019], who study the properties of instrument rules in a incomplete

markets model with a portfolio of assets including bonds and equities.12

Additionally, our study is naturally related to recent literature that emphasizes the role of

pecuniary externalities under collateral constraints, financial accelerator (balance-sheet) e§ects

and over- and underborrowing relative to the constrained-e¢cient allocation (see Benigno et

al. [2010]; Bianchi [2011]; Bianchi and Mendoza [2010]; Brunnermeier and Sannikov [2015];

Costinot et al. [2015]; Dávila and Korinek [2018]; Jeanne and Korinek [2010]; and Lorenzoni

[2008], among others).13 Devereux and Yu [2016] characterize optimal monetary policy under

discretion in a small open economy with occasionally binding borrowing constraints. Relative

to these papers, a distinct feature is our specific focus on monetary policy in a global equilib-

rium characterized by overborrowing (and obviously underborrowing in the other country) with

9The result also holds when ERPT is asymmetric across borders–the case of DCP recently emphasized byGopinath [2016]. Casas et al. [2016] study optimal monetary policy for this case, focusing on a small openeconomy.10Other contributions have looked at similar issues in a small open economy framework–see e.g. De Paoli

[2009] and Fanelli [2019].11Tille [2005] assesses the welfare impact of integrating international asset markets with nominal rigidities and

a stochastic component in monetary policy.12A number of other papers numerically solve open economy models under incomplete markets, and examine

optimal policy often using ad hoc loss functions. See, for example, Kollmann [2002].13Cavallino [2016] examines foreign exchange interventions as a second instrument (in addition to conventional

interest rate policy) available to the central bank to redress ine¢cient capital flows in an economy with borrowingconstraints similar to those of Gabaix and Maggiori [2015].

5

respect to both the first-best and the constrained-e¢cient allocation.14

Finally, as regards the debate on the limits of monetary policy, our results are in line

with Woodford [2009], showing that openness to foreign capital does not compromise monetary

control, i.e., the ability of the central bank to pursue a desired monetary stance. Yet, as stressed

by Rey [2013] and Farhi and Werning [2014], ine¢cient capital flows may create adverse trade-

o§s across policy goals, hampering a central bank’s ability to maintain the economy on an

e¢cient path. We complement these papers in that we inspect the monetary policy trade-o§s

created by capital flows, and characterize the optimal monetary response.

The rest of the paper is organized as follows. The next section briefly goes over the standard

two-good, two-country, New Keynesian model that we take as the framework for our analysis.

Section 3 derives the global loss function, discussing each of its arguments in some detail, and

characterizes the cooperative optimal targeting rules under PCP and LCP. In this section, we

also analyze in detail how and why incomplete markets make a di§erence for monetary policy.

In Section 4, we consider a baseline specification of the model that we dub the Cole and Obstfeld

(CO) economy, where capital flows are exogenous to policy and independent of ERPT. We can

therefore focus sharply on how the optimal monetary stance changes across LCP and PCP

economies. In section 5, we go beyond the role of ERPT, and further study how the optimal

monetary policy varies systematically depending on the equilibrium link between misalignment

and capital flows. Section 6 concludes. The appendix derives the loss function, the targeting

rules, and the di§erent allocations shown throughout the papers, and provides proofs for the

propositions and lemmas stated in the text.

2 The model economy

The analysis builds on the standard open economy version of the workhorse model in monetary

economics (see, e.g., Clarida, Galí and Gertler [2002] and Engel [2011]), with well-known char-

acteristics. The world economy consists of two countries of equal size, H and F . Each country

specializes in one type of tradable good, produced in a number of varieties or brands defined

over a continuum of unit mass. Brands of tradable goods are indexed by h 2 [0, 1] in the Homecountry and f 2 [0, 1] in the Foreign country. Firms producing the goods are monopolisticsuppliers of one brand only and use labor as the only input to production. These firms set

prices either in local or producer currency units and in a staggered fashion as in Calvo [1983].

Asset markets are complete at the national level, but incomplete internationally.

In what follows, we describe our setup focusing on the Home country, with the understanding

that similar expressions also characterize the Foreign economy–variables referring to Foreign

firms and households are marked with an asterisk.14Key to our results is that, in equilibrium, the natural borrowing constraints in a bond economy depend on

real exchange rate misalignment. Exchange rate movements drive di§erences in national wealth by a§ecting therelative value of a country’s output (and thus the natural constraint on foreign borrowing), similarly to theirvaluation e§ects on outstanding foreign assets and liabilities already stressed by the literature (see, e.g., Gour-inchas and Rey 2014). Since the relative value of output (and its present discounted value) reflect misalignmentwhen financial markets are incomplete, real exchange rate movements induce an ine¢cient wealth wedge acrosscountries.

6

2.1 The household’s problem

2.1.1 Preferences

We consider a cashless economy in which the representative Home agent maximizes the expected

value of her lifetime utility, where instantaneous utility U is a function of a consumption index,

C, and (negatively) of labor e§ort L, specialized as follows:

U [Ct, Lt] = ζC,tC1−σt

1− σ− κ

L1+η

1 + η, σ, η > 0 (1)

whereas the model also allows for shocks to marginal utilities of consumption ζC,t. Foreign

agents’ preferences are symmetrically defined. Households consume both domestically produced

and imported goods. We define Ct(h) as the Home agent’s consumption as of time t of the Home

good h; similarly, Ct(f) is the Home agent’s consumption of the imported good f . We assume

that each good h (or f) is an an imperfect substitute for all other goods’ varieties, with constant

elasticity of substitution θ > 1:

CH,t ≡[Z 1

0Ct(h)

θ−1θ dh

] θθ−1

, CF,t ≡[Z 1

0Ct(f)

θ−1θ df

] θθ−1

. (2)

The full consumption basket, Ct, in each country, aggregates Home and Foreign goods

according to the following standard CES function:

Ct ≡ha1/φH CH,t

φ−1φ + a

1/φF CF,t

φ−1φ

i φφ−1

, φ > 0, (3)

where aH and aF are the weights on the consumption of Home and Foreign traded goods,

respectively, and φ is the constant (trade) elasticity of substitution between CH,t and CF,t.

2.1.2 Price indexes

The price index of the Home goods is given by:

PH,t =

[Z 1

0Pt(h)

1−θdh

] 11−θ

, (4)

and the price index associated with the consumption basket, Ct, is:

Pt =haHP

1−φH,t + aFP

1−φF,t

i 11−φ

. (5)

Let Et denote the Home nominal exchange rate, expressed in units of Home currency per unitof Foreign currency. The real exchange rate (RER) is customarily defined as the ratio of CPIs

expressed in the same currency, i.e., Qt =EtP∗tPt

. The terms of trade (TOT) are instead defined

as the relative price of domestic imports in terms of exports: Tt =PF,tEtP ∗H,t

if firms set prices in

local currency andEtP ∗F,tPH,t

under producer currency pricing.

7

2.1.3 Budget constraints

Home and Foreign agents trade an international bond, BH, which pays in units of Home currency

and is zero in net supply. Households derive income from working, wtLt, from domestic firms’

profits, Π(h), lump-sum transfers Tt, and from interest payments, (1 + it)BH,t, where it is the

nominal bond’s yield, paid at the beginning of period t but known at time t−1. Households usetheir disposable income to consume and invest in bonds. The individual flow budget constraint

for the representative agent j in the Home country is therefore:

PH,tCH,t + PF,tCF,t +BH,t+1 ≤ wtLt + (1 + it−1)BH,t +Z 1

0Π(h)dh+ Tt. (6)

The household’s problem thus consists of maximizing lifetime utility, defined by (1), subject to

the constraint (6).

2.2 Firms

Firms employ domestic labor to produce a di§erentiated product h according to the following

linear production function:

Y (h) = ζY L (h) , (7)

where L (h) is the demand for labor by the producer of the good h and ζY is a technology shock

common to all producers in the Home country, which follows a statistical process to be specified

below.

Firms are subject to nominal rigidities à la Calvo so that, at any time t, they keep their

price fixed with probability α. We assume that when firms update their prices, they do so

simultaneously in the Home and Foreign markets. Following the literature, we consider two

models of nominal price distortions in the export markets. According to the first model, firms

set prices in the currency of the destination (local) market – this is the LCP hypothesis. The

maximization problem is then as follows:

MaxP(h),P∗(h) Et

(1X

k=0

pbt,t+kαk

[Pt(h)Dt+k(h) + EtP∗t (h)D∗t+k(h)

]−

MCt+k(h)[Dt+k(h) +D

∗t+k(h)

]!)

(8)

where pbt,t+k is the firm’s stochastic nominal discount factor between t and t+k, and the firm’s

demand at Home and abroad is given by:

Dt(h) =

Z (Pt(h)PH,t

)−θCH,tdh

D∗t (h) =

Z P∗t (h)P ∗H,t

!−θC∗H,tdh

In these expressions, PH,t and P ∗H,t denote the price index of Home goods in the Home and

Foreign countries – the latter expressed in Foreign currency.

By the first-order condition of the producer’s problem, the optimal price Pt(h) in domestic

8

currency charged to domestic customers is:

Pt(h) =θ

θ − 1

Et

1X

k=0

αkpbt,t+kDt+k(h)MCt+k(h)

Et

1X

k=0

αkpbt,t+kDt+k(h)

; (9)

while the price (in foreign currency) charged to customers in the Foreign country is:

P∗t (h) =θ

θ − 1

Et

1X

k=0

αkpbt,t+kD∗t+k(h)MCt+k(h)

Et

1X

k=0

αkpbt,t+kEt+kD∗t+k(h). (10)

According to the alternative model, we posit that firms set prices in the producer currency

– this is the PCP hypothesis. In this case, exchange rate pass-through is complete. Given that

demand elasticities are assumed to be the same across markets, in domestic currency the price

charged to foreign consumers is the same as the optimal price charged at Home: the law of one

price holds: P∗t (h) = Pt(h)/Et. The optimal price is similar to (9), whereas Home demand isreplaced by global demand.

Since all the producers that can choose their price set it to the same value, we obtain the

following equations for PH,t and P ∗H,t

P 1−θH,t = αP 1−θH,t−1 + (1− α)Pt(h)1−θ, (11)

P ∗1−θH,t = αP ∗1−θH,t−1 + (1− α)P∗t (h)

1−θ.

Similar relations hold for the Foreign firms.

2.3 Asset markets and exchange rate determination

In specifying the asset market structure, we restrict trade to one financial instrument only, a

safe nominal bond. While capturing the notion that international financial markets do not pro-

vide e¢cient risk insurance against all shocks, intertemporal trade still implies forward-looking

exchange rate determination, as a by-product of equilibrium in financial markets. Namely, by

combining the Euler equations for the Home households

UC(Ct, ζC,t

)

Pt= (1 + it)Et

"βUC(Ct+1, ζC,t+1

)

Pt+1

#

and the Foreign households:

UC(C∗t , ζ

∗C,t

)

P∗t= (1 + i∗t )Et

"βUC(C∗t+1, ζ

∗C,t+1

)

P∗t+1

#,

UC(C∗t , ζ

∗C,t

)

EtP∗t= (1 + it)Et

"βUC(C∗t+1, ζ

∗C,t+1

)

Et+1P∗t+1

#;

9

e¢cient trade in the international bond will imply the following uncovered interest parity con-

dition, which equates the nominal stochastic discount rates in expectations:

Et

"βUC(Ct+1, ζC,t+1

)

UC(Ct, ζC,t

) PtPt+1

#= Et

"βUC(C∗t+1, ζ

∗C,t+1

)

UC(C∗t , ζ

∗C,t

) EtP∗tEt+1P∗t+1

#(12)

Solved forward, this equation pins down the equilibrium exchange rate.

Under complete markets, the condition (12) holds state-by-state, rather than in expectations,

since agents trade in contingent assets up to the point when, at the margin, the valuation of an

extra unit of money of currency is equalized across borders in all circumstances. When countries

are symmetric, this implies that the relative utility value of wealth, denoted by Wt,

Wt ≡UC(C∗t , ζ

∗C,t

)1

EtP∗t

UC(Ct, ζC,t

)1Pt

=UC(C∗t , ζ

∗C,t

)

UC(Ct, ζC,t

) 1Qt

(13)

is identically equal to one (see, e.g., Gravelle and Rees [1992], Backus and Smith [1993] and

Obstfeld and Rogo§ [2001]). Note that the marginal utility of consumption across borders is

adjusted for the respective prices of the consumption basket.

Under incomplete markets, however, the equilibrium condition (12) only holds in expecta-

tions: any shocks will induce a wedge in the (ex post) relative value of wealth across borders,

so that in general Wt 6= 1. As shown below, Wt defines a theoretically grounded and e¢cient

measure of cross-border imbalances that arise due to asset markets imperfections in the policy

problem–in line with the approach by Woodford [2010], who studies monetary trade-o§s under

financial frictions in a closed economy setting allowing for agent heterogeneity.

2.4 Log-linearized equilibrium

Throughout the paper, the model’s equilibrium conditions and constraints will be written out in

log-deviations from the non-stochastic steady state–we will assume a symmetric steady-state

in which the net foreign asset position is zero and the markup distortion is eliminated with

appropriate subsidies. Details on the log-linearized model equations are given in appendix.

Notation-wise, we denote steady-state values of variable with an upper bar, and write bxt =lnxt/x for deviations from steady state under sticky prices. While we will study di§erent

specifications of the model–PCP vs. LCP, with either unitary or generic trade elasticity– we

will not denote variables di§erently across them, since each specification will be discussed in a

separate section or subsection. We make two exceptions to this notation convention. First, we

will use the superscript fb to denote variables in the unique “first-best” allocation, corresponding

to the case of complete asset markets, flexible prices and no markup distortions. Second, in

Sections 4 and 5, we will use the superscript na to denote variables in the “natural” (flex-price)

allocation.

Before delving into the analysis, it is useful to characterize upfront the first-best allocation,

against which we will define our loss functions and the optimal policy rules, and discuss two

key properties of the model under incomplete markets.

10

2.4.1 The first-best allocation benchmark

The first-best output in the Home and Foreign country, bY fbH,t and bYfbF,t, together with the real

exchange rate and the terms of trade are shown in Table 1.

Table 1. The first-best allocation

bY fbH,t =2aH(1−aH)(σφ−1)

(bT fbt

)−(1−aH)

(bζC,t−bζ

∗C,t

)+bζC,t+(1+η)bζY,t

η+σ

bY fbF,t =2aH(1−aH)(σφ−1)

(−bT fbt

)+(1−aH)

(bζC,t−bζ

∗C,t

)+bζ∗C,t+(1+η)bζ

∗Y,t

η+σ .

bQfbt = (2aH − 1) bT fbt = σ(bCfbt − bC∗fbt

).

bT fbt =σ(bY fbH,t−bY

fbF,t

)−(2aH−1)

(bζC,t−bζ

∗C,t

)

4(1−aH)aH(σφ−1)+1,

The table highlights a key feature of the first-best allocation, that we will extensively use in

our analysis. Even though households are forward looking, in equilibrium relative prices and

quantities depend only on the current-period (exogenous) fundamentals, not on their expected

realizations in the future–in line with the well-known results in Barro and King [1984].15 A

notable implication is that, in the first best, neither the short-term real interest rate (given by the

growth rates in marginal utility), nor the long-term interest rate (equal to current consumption)

depends on anticipated shocks.

The same applies to cross-border capital flows. To represent these flows in the e¢cient

economy, with slight abuse of notation we denote by bBfbt the “notional” real net foreign assets

in the first best, simply defined as cumulated real net exports (consumption minus income).

Furthermore, we scale real net foreign assets with steady-state output, so that bBfbt 'Bfbt − B

Yfb

.

The cross-border e¢cient financial flows, characterized up to first order, can then be written as:

bBfbt − β−1 bBfbt−1 = (1− aH)σ−1h(2aH (σφ− 1) + 1− σ) bT fbt −

(bζC,t − bζ

∗C,t

)i(14)

Only contemporaneous shocks appear on the right hand side of this expression. Thus, relative

to this benchmark, any cross-border flow of capital that may respond to anticipated future

changes in fundamentals (or news shocks) under incomplete markets is entirely ine¢cient.

2.4.2 Two key properties of the incomplete-market model

Under the model specification assuming trade in one noncontingent bond, a key property of the

log-linearized equilibrium is that, by the uncovered interest parity condition (12), cWt follows a

random walk:

EtcWt+1 = cWt. (15)

Because of incomplete risk sharing, shocks will generally result in a unit root in the relative

value of wealth across borders–corresponding to a unit root in net foreign assets. A comment

is in order in this respect. In the text to follow, we will carry out our analysis of the bond15Recall that in the workhorse monetary model we use in our analysis, preferences are time separable and there

is no capital accumulation (see Devereux and Engel [2007] for an analysis of the optimal monetary response tonews shocks under complete markets). Introducing capital accumulation and other sources of sluggish adjustment,such has habits or adjustment costs would change the results to follow mainly quantitatively.

11

economy allowing cWt (and net foreign wealth) to be not stationary. This is a choice motivated

by tractability and analytical transparency, but we will return to this point in subsection 4.1.1.

Finally, we emphasize that, under our assumption that the initial steady state is symmetric

with zero net foreign wealth, up to first order, the dynamic of net foreign assets (and thus cWt)

does not respond to the ex post returns on internationally traded bonds. In other words, real

net foreign assets are always capitalized at the steady-state real interest rate β−1. This feature

has one important implication for optimal monetary policy. Namely, starting from a symmetric

steady state with zero net foreign wealth, monetary policy cannot correct misallocations in

demand and misalignment by manipulating the ex post return on outstanding bonds to a§ect the

wealth distribution (as in, e.g., Devereux and Sutherland [2008] and Benigno [2009]). Instead,

it will have to operate by a§ecting relative prices, output and net foreign assets accumulation.

3 Why and how do incomplete markets a§ect monetary policy?

Our main objective is to examine the monetary policy trade-o§s brought about by ine¢cient

capital flows in economies where asset markets are incomplete. In this section, we first dis-

cuss the welfare-relevant gaps shaping policy trade-o§s in open economies, and reconsider how

incomplete markets a§ects the monetary transmission to macroeconomic variables. We then

derive a general quadratic policy loss function obtained from a second-order approximation of

agents’ utility for generic incomplete markets (i.e., without specifying the form of market in-

completeness). Finally, we characterize the optimal cooperative policy under commitment, in

terms of optimal targeting rules.

In an open economy, in addition to output gaps and inflation rates, both the welfare function

and targeting rules can be written as a function of real exchange rate misalignment and relative

demand misallocation, themselves a function of ine¢cient capital flows. Di§erent from the case

of complete markets, where misalignment and demand misallocation are proportional to each

other, these distortions, combined, define a gap specific to imperfect risk sharing, which we dub

“wealth gap” (directly related to Wt). We will show that, as a direct synthetic measure of the

distortions associated with ine¢cient flows, this gap plays a key role in optimal policy design,

acting much like an endogenous “markup” shock–i.e., it gives rise to meaningful trade-o§s

between inflation, output gaps, demand misallocation and misalignment.

3.1 Welfare-relevant gaps in an open economy

A recurrent theme in policy debates concerns the possibility that international relative prices

are misaligned and cross-border borrowing/lending is too high or too low–corresponding to

either excessive or insu¢cient demand in di§erent countries. Drawing on previous work of ours

(Corsetti et al. [2010]), we now define gaps to account for these policy concerns, using the same

logic underlying the definition of the welfare-relevant output gap.

As is customary in monetary stabilization analysis, we will write policy objectives and

targeting rules in terms of welfare-relevant gaps (all denoted with a tilde), expressing relevant

variables as deviations from their first-best allocation values.

12

3.1.1 Misalignment: real exchange rate gaps

We start with three relative price gaps that may open per e§ect of either nominal rigidities

or financial frictions, or both. To wit: exchange rates are misaligned when they deviate from

the value they would take in the e¢cient allocation.16 Since there are di§erent measures of

international relative prices, there are di§erent (complementary) measures of misalignment.

For the relative price of consumption across countries, the welfare-relevant gap is:

eQt = bQt − bQfbt . (16)

Analogously, for the relative price of tradables, the terms-of-trade gap is:

eTt = bTt − bT fbt . (17)

Finally, misalignment can also arise when nominal rigidities in local currency translate into cross-

border deviations from the law of one price (henceforth LOOP). In this case, identical goods are

ine¢ciently traded at di§erent prices domestically and abroad. These price di§erences define

another dimension of misalignment, which, measured on average for the basket of Home goods,

is:e∆H,t = (bEt + bP ∗H,t − bPH,t) (18)

where e∆H,t is equal to zero when the LOOP holds. Note that, to the extent that P ∗H,t and PH,tare sticky, the law of one price is violated with any movement in the exchange rate. Specifically,

domestic currency depreciation tends to increase the Home firms’ receipts in Home currency

from selling goods abroad, relative to the Home market: Home currency depreciation raisese∆H,t. Similar considerations apply to e∆F,t.

3.1.2 Demand misallocation: demand and wealth gaps

Ine¢cient external positions could be captured by tracing capital flows in excess of the financial

flows in an e¢cient allocation, i.e., bBt − bBfbt , a gap that opens in the presence of both nominaland real (financial) distortions.17 However, there are better and more informative measures.

A first one is the “relative demand gap,” denoted by eDt and defined as the cross-countrydi§erence in private (consumption) demand relative to the first best:

eDt =fCt − eC∗t .

As stressed by Engel [2011] and Fahri and Werning [2016], this gap may open also in complete

market economies, per e§ect of nominal distortions. An expression from the demand gap under

incomplete markets can be derived by taking the di§erence in budget constraints (where we

16We stress that, conceptually, the first-best exchange rate is not necessarily (and in general will not be)identical to the “equilibrium exchange rate,” traditionally studied by international and policy institutions, asa guide to policy-making. The e¢cient exchange rate is theoretically and conceptually defined, at any timehorizon, in relation to a hypothetical economy in which all prices are flexible and markets are complete. Infact, our measure of misalignment (as the di§erence between current exchange rates and the e¢cient one) isconstructed, in strict analogy to the notion of a welfare-relevant output gap, as the di§erence between currentoutput and the e¢cient level of output, which does not coincide with the natural rate (i.e., the level of outputwith flexible prices).17 It is worth stressing that this measure would be well defined also under financial autarky, whereas bBt = 0.

13

scale real net foreign assets with steady-state output, Bt =BH,t+1Pt

):

σ eDt = σh−2β−1

(eBt − β eBt−1

)i+ σ

heYH,t − eYF,t − 2 (1− aH) eTt

i(19)

+(1− aH) [4aH (1− aH) (σφ− 1)− 1]σ−1h(2aH (σφ− 1) + 1− σ) eT fbt −

(bζC,t − bζ

∗C,t

)i,

Everything else equal, capital inflows ( eBt < 0) tend to open a positive demand gap ( eDt > 0)

but in general equilibrium the response of the demand gap to the shocks driving the capital

flows will also depend on the endogenous response of the terms of trade and output gaps.

Combined with the real exchange rate gap, eQt, eDt adds up to a gap that opens only in thepresence of financial frictions (whether or not there are nominal rigidities). We define this as

the “wealth” gap, fWt:fWt = σ eDt − eQt, (20)

where fWt is equal to log-deviations in the relative value of wealth (13). If markets are complete,fWt = 0 always, even when the overall allocation is not e¢cient because of nominal rigidities or

other distortions. If markets are incomplete, instead, fWt will generally not be zero, and can have

either sign, with a straightforward interpretation. A positive gap fWt > 0 means that, given the

relative price of consumption, the consumption of the Home (national representative) individual

is ine¢ciently high vis-à-vis foreign consumption. While consumption smoothing is optimal

from an individual-agent perspective in response to anticipated shocks, from a global welfare

perspective relative Home wealth would be too high.18 Conversely, a negative gap suggests that

relative Home demand is ine¢ciently low given the exchange rate, and/or, for a given eDt, theshock causes ine¢cient real depreciation (relative to first best).

3.2 The wealth gap and monetary policy trade-o§s specific to incompletemarkets

The wealth gap defined in the previous subsection fully captures the implications of imper-

fect financial markets for the policy trade-o§s faced by policy-makers in the design of optimal

stabilization rules. Under complete markets, the demand gap eDt and real exchange rate mis-alignment eQt can each be di§erent from zero–depending on the e§ect of nominal rigidities or

other distortions (e.g., taxes or markup shocks). Yet, as a consequence of full risk sharing,

they will always remain proportional to each other: fWt = σ eDt − eQt = 0. Closing eQt will betantamount to closing eDt, and vice versa. Under incomplete markets, instead, since fWt will

generally deviate from zero, eDt and eQt are no longer proportional to each other. In general, theoptimal monetary rule will not close any of these gaps completely, but will have to minimize

these gaps jointly with inflation and output gaps.

The wealth gap itself confronts monetary authorities with a fundamental trade-o§. A mon-

etary easing leans against real over-appreciation eQt, which per se reduces the wealth gap;however, by stimulating a domestic demand boom, it also raises eDt, which increases the wealth18With incomplete markets, price movements are not e¢cient. An appreciation of the real exchange rate as-

sociated with a Home consumption boom is a leading example of a pecuniary externality. While fully rationalfrom an individual perspective, agents’s decisions to borrow and lend move international relative prices ine¢-ciently. These are no longer correct indicators of relative scarcity: consumption is higher where the price of theconsumption bundle is also higher; see Geanakoplos and Polemarchakis [1986].

14

gap. In some notable cases (which we analyze in detail in Section 4), the wealth gap fWt (and

the associated capital flows) is exogenous to policy, so that these two e§ects must exactly o§set

each other. When this is case, monetary authorities will not be able to a§ect the combined

ine¢ciencies arising from both the misallocation in demand and the real exchange rate mis-

alignment, regardless of LCP and PCP. Monetary policy may nonetheless a§ect their relative

size.19

When fWt (and thus capital flows) depends on monetary policy, its response to a monetary

expansion depends on structural features such as risk aversion σ, the trade elasticity φ, the

degree of openness aH (under the maintained assumptions of home bias, aH ≥ 1/2), and on

nominal price rigidities α. The following proposition states threshold values of the trade elasticity

as a function of σ, aH and α, above which expansionary monetary policy always widens fWt, that

is, the e§ect of an expansion on relative demand always prevails on its e§ect on the exchange rate.

In addition, the proposition states that, while widening fWt, a monetary easing also decreases

ine¢cient capital inflows. The threshold di§ers across PCP and LCP economies.

Proposition 1: Under the maintained assumptions of home bias ( aH ≥ 1/2) and linear

disutility of labor ( η = 0), monetary easing always widens fWt (> 0) but decreases ine¢cient

capital inflows ( eBt > 0) for a trade elasticity φ above the following threshold under PCP

φ >1 + 2aH−1

σ

2aH> 0;

and, for σ > 1, φ ≥ 1 under LCP.Proof. See the appendix.!The wealth gap has substantial implications for inflation dynamics. As discussed at length

in previous work of ours (CDL 2010), given consumption, equilibrium nominal wages and thus

marginal costs respond to imported inflation hence to exchange rate misalignment. Given

misalignment, nominal wages increase with equilibrium consumption, in turn a function of

borrowing and financial flows, hence of the wealth gap. Below we write out the Phillips Curves

(four of them under LCP, collapsing into two under PCP), as a function of misalignment and

wealth gaps (whereas we used the fact that, under symmetry, e∆H,t = e∆F,t = e∆t, see Engel[2011]):

πH,t − βEtπH,t+1 = (21)

(1− αβ) (1− α)α

"(σ + η) eYH,t

− (1− aH)h2aH (σφ− 1)

(eTt + e∆t

)− e∆t − fWt

i#

π∗H,t − βEtπ∗H,t+1 = πH,t − βEtπH,t+1 +

(1− αβ) (1− α)α

b∆t,

19As shown in Section 4, monetary policy will be able to determine in a constrained-e¢cient way how to spreadthe welfare costs of macroeconomic adjustment across the di§erent gaps, including the two components of fWt.

15

π∗F,t − βEtπ∗F,t+1 =

(1− αβ) (1− α)α

"(σ + η) eYF,t

(1− aH)h2aH (σφ− 1)

(eTt + e∆t

)− e∆t − fWt

i#

πF,t − βEtπF,t+1 = π∗F,t − βEtπ∗F,t+1 −

(1− αβ) (1− α)α

e∆t,

By inspecting the expressions above, it is apparent that the wealth gap is isomorphic to inef-

ficient markup shocks, typically included in the analysis of the Phillips Curves–see e.g. our

previous work CDL [2010]. With incomplete markets, misalignment and imbalances naturally

create a trade-o§ between inflation and unemployment, without the need of assuming exogenous

cost-push disturbances.20

3.3 A general (quadratic) global policy loss function

From the model, we derive a second-order approximation of the equally weighted sum of the

utility of the Home and Foreign national representative agents–written in terms of the gaps

defined above, all in quadratic forms. As stated in the proposition to follow, in open economy,

the policy loss functions include both “internal” objectives (inflation and output gaps), and

“external” ones (relative price misalignment and the relative demand gap). Our contribution

is to bring modern monetary theory to bear on these traditional categories, with a precise

theory-consistent reformulation of these objectives as arguments in the loss function.

Proposition 2: Under the assumption that appropriate subsidies o§set firms’ markups todeliver an e¢cient, non-distorted steady state, the period-by-period quadratic welfare function

for incomplete market economies is as follows:

LWt −(LWt)fb n (22)

−1

2

8><

>:

(σ + η)(eY 2H,t + eY 2F,t

)+

α

(1− αβ) (1− α)θ(π2t + π

∗2t

)

−2aH (1− aH)

4aH (1− aH) (σφ− 1) + 1

[(σφ− 1)σ

(eYH,t − eYF,t

)2− φ

(e∆t + fWt

)2]

9>=

>;

+t.i.p.,

Proof. See the appendix.!In writing the above loss, for convenience, we have substituted out the terms-of-trade mis-

alignment using its equilibrium relation with output gaps, deviations from the law of one price,

and relative demand gaps. Observe that the expression is written in terms of CPI inflation and

includes deviations from the LOOP, so that it directly applies to the LCP economy. Yet, its

PCP counterpart can be readily obtained by setting the LOOP deviations to zero (e∆t = 0), andusing the fact that, under the law of one price, the inflation term π2t ≡ aHπ2H,t + (1− aH)π

2F,t

and π∗2t ≡ aHπ∗2F,t + (1− aH)π∗2H,t reduces to π

2t ≡ π2H,t and π

∗2t ≡ π∗2F,t.

21

20When markets are incomplete, the distinction between “e¢cient” and “ine¢cient” shocks usually drawn byboth the closed-economy literature and the open economy literature assuming perfect risk sharing becomes lessuseful for the purpose of policy design. Also shocks to tastes and technology (labelled “e¢cient”) endogenouslyopen a wealth gap and create misalignments–and thus raise meaningful policy trade-o§s between output andinflation under both LCP and PCP.21Similarly, in related work we show that the loss-function under the case of asymmetric ERPT with DCP,

stressed by Gopinath [2016], is a particular case of the above loss-function under symmetric LCP.

16

The expression (22) encompasses the cases of financial autarky (no asset is traded interna-

tionally), international trade in one bond, as well as international trade in any number of assets,

including complete markets. In this sense, the above loss function generalizes and complements

the ones derived in previous work of ours (CDL [2010]) for the case of autarky and complete

markets.22

3.4 Optimal targeting rules in bond economies

To characterize the optimal cooperative policy under commitment, we focus on economies in

which the only asset traded across border is a non-contingent, nominal bond–under our as-

sumption of zero Net Foreign Asset in steady state, the bond currency denomination is not

relevant for the policy problem. The derivation of the targeting rules, while complex, is stan-

dard: we maximize the present discounted value of the sum of (22) over time, subject to the

log-linearized equilibrium conditions and constraints characterizing the competitive equilibrium

allocation in bond economies. In the interest of transparency and tractability, we adopt a

timeless perspective (see, e.g., Woodford [2010]).

Following a common practice in international economics, we synthesize the optimal coop-

erative policy in terms of two targeting rules: a global rule summing up inflation and output

gaps across countries, and a cross-country rule, expressed in terms of di§erences in gaps across

countries. These are presented in the propositions 3 through 6 below.

Proposition 3: From a global perspective, the optimal targeting rule under cooperation and

commitment is given by

0 =(eYH,t − eYH,t−1

)+(eYF,t − eYF,t−1

)+ (23)

θ[aHπH,t + (1− aH)πF,t + aHπ∗F,t + (1− aH)π

∗H,t

],

where in the case of a PCP economy the inflation term becomes πH,t + π∗F,t – since, under

PCP, world CPI and PPI inflation rates coincide.

Proof. See the appendix.!From a global perspective, the optimal cooperative monetary policy stabilizes output gaps

and inflation at the global level. To the extent that world inflation is zero (in the absence of

exogenous markup shocks), the sum of output gaps and consumption deviations is also zero. An

important implication is that the optimal monetary stance will have the opposite sign across

countries.23

Deriving cross-country or country-specific rules involves solving a system of di§erence equa-

tions in the di§erent Lagrange multipliers from the optimal policy problem, which di§er across

LCP and PCP economies. In the LCP case, tractable general expressions–comparable to the

global rule–can be derived only under some parameter restrictions. We will thus analyze the

LCP and PCP economies in turn.22Gaps (other than output gaps and inflation) similar to the ones we use in our analysis identify policy objectives

arising from heterogeneity among sectors and agents in economies distorted by financial imperfections, in additionto nominal rigidities (see, e.g., Cúrdia and Woodford [2016] for an analysis in a closed economy).23Another implication is that we can write eDt ≡ eCt − eC∗t = 2 eCt. These results also hold in the natural rate

allocation.

17

3.4.1 Incomplete pass-through (LCP) economies

In the LCP case, a tractable rule is derived by Engel [2011] under the assumptions that markets

are complete and labor elasticity is infinite (η = 0). An important result in our paper is that,

as long as η = 0, it is possible to derive a tractable cross-country targeting rule also under

incomplete markets. This is given by the following proposition:

Proposition 4: Under LCP, if η = 0, the optimal policy under cooperation and com-

mitment is fully characterized by the general global rule (23) and the following cross-country

(di§erence) rule:

0 = θ (πt − π∗t ) + eDt − eDt−1 (24)

+4aH (1− aH)φ2aH (φ− 1) + 1

(σ − 1)σ

h(fWt − fWt−1

)+(e∆t − e∆t−1

)i,

where aHπH,t + (1− aH)πF,t = πt and (1− aH)π∗H,t + aHπ∗F,t = π

∗t .

Proof. See the appendix.!It is worth noting that the cross country rule under complete markets is given by the first

two terms on the right hand side of (24), with the CPI inflation and consumption di§erentials

as the only arguments. The last term in (24), in the wealth gap and deviations from the law of

one price, is specific to incomplete markets economies.

An important property of LCP economies under incomplete markets (somehow missed by

the literature so far) allows us to derive a simpler version of the above rule. Namely, a key

result derived by Engel [2011] under complete markets is that, as long as η = 0, the relative

prices eTt + e∆t are exogenous with respect to monetary policy–for any value of σ. As stated inthe following proposition, the same result also holds under incomplete markets, provided agents

have log-utility, i.e., σ = 1. In addition, the proposition states an additional, important result:

under the same parameterization, both capital flows and the wealth gap fWt are also una§ected

by monetary policy.24

Proposition 5. In LCP bond economies, as long as η = 0 and σ = 1, relative priceseTt + e∆t, cross-border capital flows ( eBt) and the wealth gap (fWt) are independent of monetary

policy for any value of trade elasticities φ.

Proof. See Appendix.!

As a corollary, focusing on the case σ = 1, we can combine the global and the cross- country

rule, so to rewrite the optimal (cooperative) policy in terms of two symmetric country-specific

rules.25

Corollary 1. In LCP bond economies, as long as η = 0, σ = 1, and absent exogenous

markup shocks, the targeting rule for the Home economy is as follows

0 = θπt + 1/2 ·h(fWt − fWt−1

)+(eQt − eQt−1

)i(25)

= θπt +(eCt − eCt−1

).

24One may observe that the last term on the right-hand side of the optimal rule (24) drops out when σ = 1:the expression for the cross-country rule (24) is the same under both complete and incomplete markets. However,as explained in the text, it does not follow that monetary policy is the same in the two cases.25Recall that absent exogenous markup shocks, global inflation and global output gaps are both zero under

the optimal policy.

18

Proof. Set σ = 1 in (24) and combine with (23).!

When markets are complete (fWt = 0), the above reduces to the expression derived by Engel

[2011]: with perfect risk insurance, provided that shocks are “e¢cient” (i.e., they a§ect tastes

and/or technology only), the optimal policy sets CPI inflation rates to zero. A zero inflation

policy closes the consumption gap and eliminates real exchange rate misalignment at once–

reflecting the fact that these gaps are proportional to (exogenous) relative prices eTt + e∆t. Thisis not possible when markets are incomplete (fWt 6= 0).

It may be worth stressing that under LCP closing the real exchange rate misalignment (i.e.,

setting eQt = 0) does not necessarily eliminate deviations from the law of one price–nor preventine¢cient deviations from the law of one price e∆t from mapping into output gap fluctuations.

This is apparent from the following expression:

(eTt + e∆t

)=eQt − e∆t2aH − 1

=σ(eYH,t − eYF,t

)− (2aH − 1)

(fWt + e∆t

)

4aH (1− aH) (σφ− 1) + 1.

Because of nominal distortions in import and export pricing in local currency, the optimal

constrained allocation cannot be first best, whether or not risk sharing is perfect.

3.4.2 Complete pass-through (PCP) economies

The analytics of the cross-country targeting rule under PCP stands in sharp contrast to the

LCP case above. No parameter restriction is required to derive a compact expression for the

following cross-country targeting rule in a bond economy.

Proposition 6: In the PCP bond-economy, the optimal policy under cooperation and com-mitment is characterized by the global rule (23) in conjunction with the following cross-country

targeting rule:

0 =h(eYH,t − eYH,t−1

)−(eYF,t − eYF,t−1

)+ θ

(πH,t − π∗F,t

)i(26)

+ 4aH(1−aH)φσ+η(4aH(1−aH)(σφ−1)+1)

2aH(σφ−1)+1−σ2aH(φ−1)+1

(fWt − fWt−1

).

which holds without the need to impose parametric restrictions on σ,η and φ.

Proof. See the appendix.!In a bond economy, the optimal cross-country targeting rule introduces a trade-o§ between

output gaps and inflation rates on the one hand, and the wealth gap on the other hand, which

is absent under complete markets–whereas (as shown by, e.g., Engel [2011] and CDL [2010])

the cross-country targeting rule is:

0 =(eYH,t − eYH,t−1

)−(eYF,t − eYF,t−1

)+ θ

(πH,t − π∗F,t

). (27)

Combining once again the global and cross-country rules for bond economies, (absent exoge-

nous markup shocks) we can write a country-specific (cooperative) rule for the Home economy:

0 =heYH,t − eYH,t−1 + θπH,t

i+ 2aH(1−aH)φ

σ+η(4aH(1−aH)(σφ−1)+1)2aH(σφ−1)+1−σ2aH(φ−1)+1

(fWt − fWt−1

).

19

from which we derive the following important corollary.26

Corollary 2: In the PCP bond-economy, if either markets are complete (fWt = 0) or

σ = φ = 1, the optimal policy can be characterized by a pair of country-specific rules, which are

a function of purely domestic objectives. For the Home country, such rule is:

eYH,t − eYH,t−1 + θπH,t = 0. (28)

Proof. Set fWt = 0 or σ = φ = 1 in (26) and combine with (23).!

Each country would stabilize its own output gap and GDP-deflator inflation–a result that

identifies a notable (and widely discussed) case of “isomorphism” of optimal policy in closed

and open economies.

4 Optimal trade-o§s and exchange rate pass-through

In this and the next section, we bring our analysis to bear on the optimal conduct of monetary

policy in economies that experience ine¢cient capital flows and study the macroeconomic dy-

namics that result from the implementation of the optimal targeting rules, contrasting LCP and

PCP. We find it convenient to present our results in two steps. First, in this section, we specify

a bond economy with log-consumption utility (σ = 1) and linear disutility of labor (η = 0)–two

restrictions motivated by tractability in the case of LCP–as well as a unitary trade elasticity

(φ = 1). Because of the latter assumption, we dub this model specification “Cole and Obstfeld”

or CO economy, after Cole and Obstfeld [1991]. In the CO economy, capital flows are exogenous

to monetary policy and independent of the specification of nominal rigidities in export pricing

(LCP or PCP). This allows us to flesh out how optimal policy changes with ERPT, holding

constant the size of the inflows. As a second step, in Section 5, we extend the analysis going be-

yond the role of ERPT, and analyze how optimal monetary policy is shaped by the equilibrium

link between capital flows and misalignment.

For the sake of analytical clarity, we will focus the analysis on shocks in the form of “news”.

In the first-best allocation the current values of macro variables do not respond at all to news

foreshadowing changes in fundamentals in the future: the response of “gaps” (in anticipation

of future changes in technology and preferences) thus coincides with the response in the equi-

librium allocation until the anticipated shock materializes–with obvious gains in tractability

and analytical transparency. An additional benefit, emphasized by Devereux and Engel [2007],

is that the analysis of “news shocks” highlights the forward-looking nature of exchange rate

determination.26For comparison, the cross-country rule under financial autarky (derived in Corsetti et al. [2010]) is as follows:

0 = [σ + η (4aH (1− aH) (σφ− 1) + 1)]nheYH,t − eYH,t−1

i−heYF,t − eYF,t−1

i+ θ

(πH,t − π∗F,t

)o+

4aH (1− aH)φ2aH (σφ− 1) + 1− σ2aH (φ− 1) + 1

(cWt − cWt−1

)+

2 (1− aH)[2aH (σφ− 1)σ − (σ − 1)

4aH (1− aH) (σφ− 1) + 12aH (φ− 1) + 1

]θ(πH,t − π∗F,t

).

20

4.1 A “Cole and Obstfeld” economy with capital flows exogenous to policy

As is well known since Cole and Obstfeld [1991] and subsequent work, in an environment with

a Cobb-Douglas aggregator of domestic and imported goods (φ = 1), log consumption utility

(σ = 1) and symmetric home bias, production risk is e¢ciently shared via endogenous terms-

of-trade movements, regardless of whether financial markets are complete or incomplete (this

applies to, e.g., productivity and markup shocks). However, full risk sharing is not granted in the

presence of other sources of risk directly a§ecting net foreign assets, ranging from political risk

(i.e., capital controls; see, e.g., Acharya and Bengui [2016]), to shocks to financial intermediation

(see, e.g., Gabaix and Maggiori [2015]) and/or preference for foreign assets (see, e.g., Cavallino

[2019]), as well as preference shocks impinging on savings. As many of these shocks have broadly

similar analytical representations, there is little or no loss of generality in focusing on shocks to

preferences that a§ect the intertemporal valuation of consumption, thus resulting in a motive

to save and lend across borders, and generating cross-country capital flows.

4.1.1 Financial flows in the first-best allocation and in the bond economy

We have already shown that, in the first-best allocation, no macro variable (but the long-term

interest rate) responds to news shocks. This is apparent in the CO economy, in which, imposing

σ = φ = 1, our (notional) measure of e¢cient capital flows (14) simplifies as follows:

bBfbt − β−1 bBfbt−1 = − (1− aH)(bζC,t − bζ

∗C,t

). (29)

In the CO economy, e¢cient inflows ( bBfbt < 0) can only be driven by relative shocks to pref-

erences in the Home country that are contemporaneous (bζC,t − bζ∗C,t > 0). Furthermore, the

assumption that η = 0 implies that news shocks will have no e§ect on exchange rates and

relative prices:27

bQfbt = (2aH − 1) bT fbt = 0

It follows that any borrowing/lending and any exchange rate movement in response to news

shocks will provide a direct measure of welfare-relevant gaps.

Specifically, compare the current account in the CO economy, given by the following expres-

sion:

bBt = bBt−1 + (1− aH)β1X

j=0

βjEt

h(bζC,t+1+j − bζ

∗C,t+1+j

)−(bζC,t+j − bζ

∗C,t+j

)i, (30)

with the notional capital flows in the first best (29). An anticipated future fall in the relative

degree of impatience(bζC,t+1+j − bζ

∗C,t+1+j < 0

)that causes capital to flow into the Home coun-

try in the bond economy (recall that a negative bBt denotes inflows into the Home country),would trigger no (notional) e¢cient flows under perfect risk sharing. Note that the size of

27With σ = φ = 1,but η > 0, Home preference shocks in favor of current consumption systematically result ina Home currency real appreciation:

bQfbt = −

η

1 + η(2aH − 1)2

(bζC,t − bζ

∗C,t

)

21

the ine¢cient borrowing and lending is increasing in openness (decreasing in home bias aH).

Ine¢cient capital flows in turn open a wealth gap:28

(1− aH)fWt = −(bBt − β−1 bBt−1

)− (1− aH)

(bζC,t − bζ

∗C,t

). (31)

The expressions (30) and (31) highlight two important properties of the CO economy. First, bothbBt, and fWt are a function of the exogenous preference shocks only, and therefore independent

of nominal rigidities and monetary policy regimes. Second, a capital inflow ( bBt < 0) driven bynews shocks will invariably lead to a positive wealth gap. As the Home economy accommodates

a higher desire to save among Foreign residents, the relative Home demand eDt grows excessively,and/or the real exchange rate becomes misaligned.29

Before proceeding, it is important to stress that the exogeneity of bBt and fWt remains un-

a§ected if cross-border flows are subject to costly intermediation in the vein of Gabaix and

Maggiori [2015] – a result emphasized by Cavallino [2019]. A simple way to capture costly

intermediation in our framework is to posit deviations from the uncovered interest rate parity

condition that are proportional to net foreign assets:

EtfWt+1 − fWt = −Γ bBt.

With this modification, the solutions for bBt and fWt become:

bBt = γ1 bBt + (1− aH)1X

j=0

γ−j−12 Et

h(bζC,t+1+j − bζ

∗C,t+1+j

)−(bζC,t+j − bζ

∗C,t+j

)i,

fWt = −

2

4(bζC,t − bζ

∗C,t

)+

1X

j=0

γ−j−12 Et

h(bζC,t+1+j − bζ

∗C,t+1+j

)−(bζC,t+j − bζ

∗C,t+j

)i−

γ1 − β(1− aH)β

bBt−1

3

5 ,

where β < γ1 < 1 < γ2.30 Both fWt and bBt are still functions of exogenous shocks only, so the

28By using (30), one can also write this expression as (1− aH)EtcWt+s =

− (1− aH)

2

6664

(bζC,t − bζ

∗C,t

)+

βP1

j=0 βjEt

2

4

(bζC,t+1+j − bζ

∗C,t+1+j

)+

−(bζC,t+j − bζ

∗C,t+j

)

3

5

3

7775+ 1−β

βbBt−1.

29From (30) and (31), it should also be clear that capital inflows are not necessarily associated to a positivewealth gap. Notably, both bBt and fWt can be negative in response to contemporaneous (as opposed to “news”)taste shocks, which raise the utility of current Home consumption (and associated with a relative increase ine¢cient output, bY fb

H,t−bYfbF,t > 0). In this case, although capital flows into the Home country, domestic consumption

is ine¢ciently low relative to the foreign one. A key di§erence between contemporaneous and news shocks topreferences is that, with the former, bBt and fWt have the same sign, while with the latter they have the oppositesign.30Specifically, γ1 and γ2 are the roots of the characteristic equation associated with the second-order di§erence

equation:

fWt =

bBt−1 − β bBt(1− aH)β

!

−(bζC,t − bζ

∗C,t

)

EtfWt+1 − fWt = −Γ bBt,

namely:βγ2 − (1 + β + βΓ) γ + 1 = 0.

22

optimal targeting rules are the same as those derived above under both LCP and PCP for the

CO economy. Clearly, setting Γ = 0 in the last expression leads to γ1 = 1 and γ2 = 1/β, which

yields expressions (30) and (31) above. We abstract from intermediation costs in the rest of our

analysis.

4.1.2 The natural rate allocation

With imperfect insurance, ine¢cient capital flows open a wealth gap and result in misallo-

cation independently of price stickiness. Table 2 shows the flexible price allocation for the

CO economy–which coincides with the natural rate allocation. In this table, all variables are

expressed as deviations from this allocation–defining gaps denoted with a superscript “na.”

Table 2. The natural rate allocation in the CO economyeY naH,t = −eY naF,t = − (1− aH)fWt

eT nat = −fWt

eQnat = − (2aH − 1)fWt

eDnat = 2 (1− aH)fWt

eCnat = − eC∗nat = 12eDnat = (1− aH)fWt

In the CO economy, under flexible prices, output gaps, exchange rate misalignment and the

relative demand gap are all proportional to the (exogenous) gap fWt. When fWt > 0 and bBt < 0,as is the case in response to news shocks, capital inflows result in a negative welfare-relevant

output gap, an overvalued real exchange rate and an excessive level of domestic consumption,

both in absolute terms and relative to Foreigners. The Foreign economy just mirrors the Home

responses–through their e§ects on fWt, the ine¢ciencies in the shock transmission are purely

redistributive. Note that, with aH > 1/2, in equilibrium, adjustment to shocks requires Home

real appreciation. Intuitively, the capital inflow into Home amounts to a transfer of purchasing

power from abroad. Since there is home bias in demand, if relative prices did not adjust, the

transfer would translate into an excess supply of Foreign goods.

In response to news shocks, as fWt > 0, all gaps widen on impact. Afterwards, since in the

linearized equilibrium EtfWt+1 = fWt, gaps remain constant.31 Note that, in the intervening

period between the arrival of the news and future changes in fundamentals, the short-term

natural rate of interest (equal to the growth rate of consumption under flexible prices) is not

a§ected at all by the shocks.32

It is well understood that, in general, an allocation with price stability under PCP is the

same as the natural rate allocation, but an allocation with CPI stability under LCP is not.

Nonetheless, it can be shown that the expressions for consumption demand and relative demand

in Table 2 hold also under CPI price stability in the LCP economy. This result will be useful

in the analysis below.

31When fundamentals change in the future, of course, macroeconomic variables will change again, includingboth deviations bCnat+s and e¢cient consumption bC

fbt+s, but not bQna

t , under η = 0.32 It follows that a monetary policy framework equating the policy rate to the short-term natural rate would

be initially unresponsive to the capital inflows.

23

4.2 Domestic demand stabilization with incomplete pass-through (LCP economies)

We now analyze the dynamics of the CO economy under the optimal policy with LCP. Below

we show the constrained-e¢cient allocation–obtained by writing out the economy dynamics in

response to a news shock when monetary authorities in each country implement the optimal

targeting rule (25). The Home allocation is shown in Table 3, as a function of the (exogenous)

wealth gap (31))–the Foreign one is the symmetric counterpart.

Table 3: Constrained-e¢cient allocation under LCP in the CO economyeYH,t = 2aH (1− aH)

(eTt + e∆t

)+ 1/2 · (2aH − 1) eDt

θπt = − (1− aH)(β{2 − 1)β{2

fWt +12

[(β{2 − 1)β{2

fWt−1 + (1− {1) eQt−1]

eTt + e∆t = −(βν2 − 1)βν2

fWt + ν1

(eTt−1 + e∆t−1

)

eQt = − (2aH − 1)(β{2 − 1)β{2

fWt −1

β{2

(fWt − fWt−1

)+ {1 eQt−1

eDt = 2 (1− aH) (β{2−1)β{2fWt +

1

β{2fWt−1 + {1 eQt−1.

In the table, {1, ν1 and {2, ν2 denote, respectively, stable and unstable eigenvalues–where theformer (ν1, {1) are increasing, the latter ({2, ν2) decreasing in the degree of price stickinessα. Observe that higher values of ν1 and {1 (corresponding to higher price stickiness) imply aslower adjustment of eTt+ e∆t as well as a slower adjustment of misalignment eQt and the demandgap eDt under the optimal policy. We state useful relations between eigenvalues in the followingLemma.33

Lemma 1. For 0 < α < 1, the variables (eigenvalues) {1, ν1 and {2, ν2 are related asfollows:

0 <(β{2 − 1)β{2

< 1, 0 <(βν2 − 1)βν2

< 1,

(β{2 − 1)β{2

>(βν2 − 1)βν2

.

The impact response to shocks and the ensuing dynamics under the optimal policy is derived

33Namely for {1,2:

{1,2 =1 + β +

(1− αβ) (1− α)α

θ ±

s[1 + β +

(1− αβ) (1− α)α

θ

]2− 4β

2β

and ν1,2 di§er from the above only in that the term(1− αβ) (1− α)

αis not multiplied by θ. As a result, we have

the following relations:

0 < {1 < 1 < β−1 +(1− αβ) (1− α)

αβθ < {2

0 < ν1 < 1 < β−1 +

(1− αβ) (1− α)αβ

< ν2,

{2 ≥ ν2It is worth noting that the eigenvalues {2 and ν2 determine the discounted value of expectations of future

fundamentals in driving the dynamics of the real exchange rate and of relative prices eTt + b∆t. Note that thelower the unstable eigenvalues {2 and ν2, the less expected future fundamentals are discounted in determiningthe gaps.

24

using the expressions in Table 3 and Lemma 1. Specifically, consider the world-economy response

to news shocks at time t0, resulting in capital inflows and a positive wedge gap fWt0 > 0. The

characterization of monetary policy follows evaluating the impact response of inflation in Table

3, that is,

πt0 = − (1− aH)(β{2 − 1)θβ{2

fWt0 ≤ 0; (32)

whereas the response of Foreign inflation is symmetric, π∗t0 = −πt0 . This establishes that, underthe optimal cooperative policy, the monetary response is contractionary and deflationary at

Home, while expansionary and inflationary abroad. Relative to a regime of strict CPI stability,

optimal policy will thus focus on stabilizing relative demand, trading-o§ this objective for

inflation and real exchange rate/misalignment variability (recall that, since bQfbt0 = 0 in responseto news shocks, the welfare relevant gap and the real exchange rate move one-to-one: eQt0 = bQt0).We summarize the key properties of the allocation in the following proposition.

Proposition 7. In the Cole and Obstfeld economy with σ = φ = 1 and η = 0, under

LCP, in response to news shocks generating ine¢cient capital flows, the real exchange rate and

CPI inflation are more volatile under the optimal policy than in a regime pursuing strict CPI

stability; on impact the relative demand gap is less volatile while the output gap is smaller.

Proof. The proof follows from Table 3 and Lemma 1, evaluating the expressions in the

table, without loss of generality, for a news shock at time t0 resulting in a Home capital inflow

and a positive wedge gap fWt0 > 0. The fact that Home CPI inflation is not stabilized and falls

on impact follows from (32).

The impact appreciation of the Home real exchange rate under the optimal policy follows

from:eQt0 = −

[(2aH − 1)

(β{2 − 1)β{2

+1

β{2

]fWt0 < 0. (33)

Since the expression in square brackets is greater than one, the impact appreciation is larger

than under CPI price stability, whereas the expression for the real exchange rate under CPI

stability coincides with eQnat = − (2aH − 1)fWt (see Section 4.1.2).34

The result that relative demand eDt0 is smaller than under strict CPI stability but stillpositive, follows from

eDnat = 2 (1− aH)fWt > eDt0 = 2 (1− aH)(β{2 − 1)β{2

fWt0 > 0 (35)

whereas the first inequality holds since(β{2 − 1)β{2

< 1, and we use the fact that the expression

for the relative demand under CPI stability coincide with eDnat .Finally, to show that the output gap is smaller than under strict CPI stability, we first

34 It is worth observing that, dynamically, the optimal stance induces a predictable exchange rate dynamic,where Home real appreciation is followed by depreciation. To illustrate this dynamic, one can use the expressionfor eQt in Table 3 to decompose the movement of the exchange rate into a long-run permanent appreciationcomponent and a component driven by the expected cumulated real interest rate di§erential across countries.Comparing the two, what determines this dynamic is the following inequality:

1

β{2> (2aH − 1)

(β{2 − 1)β{2

{1(1− {1)

= (2aH − 1)1

β{2. (34)

The expected appreciation in the long run reflects the permanent wealth e§ects associated with the capital inflowunder incomplete markets.

25

rewrite the expression in Table 3 as follows:

eYH,t0 = 2aH (1− aH)(eTt0 + e∆t0

)+ 1/2 · (2aH − 1) eDt0

= (1− aH)[2aH

((β{2 − 1)β{2

−(βν2 − 1)βν2

)−(β{2 − 1)β{2

]fWt0 ;

and derive the output gap under strict CPI stability, eY CPIH,t0, given by:

eY CPIH,t0 = (1− aH)[2aH

(1−

(βν2 − 1)βν2

)− 1]fWt0 .

The result directly follows from comparing the two expressions using Lemma 1. Observe that,

by Lemma 1 and since ν2 < {2, neither the output gap under the optimal policy nor eY CPIH,t0is

necessarily negative (i.e., eYH,t0 Q 0, and eY CPIH,t0Q 0).!

The proposition establishes that in response to a capital inflow, the (constrained-) optimal

contractionary stance at Home matched by the expansion abroad contains the ine¢cient surge in

Home consumption relative to the Foreign one. However, concerns about inflation stabilization

implies the cooperative policy falls short of closing the demand gap. Moreover, the Home

output gap is always lower than under strict CPI stabilization, though not necessarily negative.

Looking at the expression for the output gap in Table 3, a positive output gap is possible if the

positive impact of the capital inflow on the relative demand gap, eDt0 outweighs the negative(and exogenous) e§ect of the terms-of-trade gap and deviations from the LOOP, eTt0 + e∆t0 . Inthis case, the volatility of the output gap is always smaller under the optimal policy than under

CPI stabilization. It is easy to see that, on impact, the output gap is positive if the following

condition is satisfied:β{2 − 1β{2βν2

− 1< 2aH.

This condition is more likely to hold in economies that are quite closed (i.e., economies with

a high home bias aH)–intuitively, openness increases the relative weight of(eTt0 + e∆t0

)and

decreases that of eDt0 in the output gap expression above. Strikingly, the inequality is al-

ways violated (for any degree of openness), in the limit case where prices are very flexible

({2 ' ν2 !1).35

Together, these results establish that in the CO economy, LCP motivates monetary author-

ities to optimally trade o§ stabilization of domestic demand (and in less open economies the

output gap) with inflation and real exchange rate volatility. We conclude this section by ex-

ploring how the policy and the dynamics change with the degree of nominal rigidities (and thus

exchange rate pass-through) and openness. As stated in the following corollary, it turns out

that both have similar implications for the equilibrium response of the exchange rate: exchange

rate volatility is higher in less open economies (larger aH) with less price stickiness (smaller α).

Corollary 3. The impact response of the real exchange rate in (33) and of the demand gap

35When cWt0 < 0–e.g. due to contemporaneous taste shocks–Home monetary policy is relatively expansionaryto stimulate the ine¢ciently low domestic consumption. Relative to the above, the response of optimal monetarypolicy is the opposite, because capital inflows are now ine¢ciently low. The real exchange rate depreciates andis undervalued. However, undervaluation is lower with a high degree of pass-through and openness.

26

in (35) are, respectively, increasing and decreasing in α > 0 and aH ≥ 1/2.

Intuitively, for a given exogenous wealth gap fWt0 , as the economy becomes less open, do-

mestic monetary policy becomes more concerned in dealing with a demand boom fueled by

capital inflows, for any given degree of price stickiness, at the expense of larger misalignment.

By the same token, if prices become stickier, implying less exchange rate pass-through, optimal

monetary policy is less concerned with redressing misalignment, since exchange rate movements

are less consequential for the domestic output gap. On the contrary, it attaches a larger weight

on aggregate demand stabilization.36

4.3 Exchange rate stabilization and competitiveness with complete pass-through (PCP economies)

A comparison of our results across LCP and PCP economies is particularly suitable in the

Cole-and-Obstfeld specification, since in response to identical shocks, the sign and size of the

ensuing capital flows and wealth gap–that is, the expressions for bBt and fWt in (31) and (30)–

are exactly the same. Conditional on a given bBt < 0 and the associated fWt (always positive

when preference shocks are anticipated), Table 4 presents the allocation under the optimal

cooperative monetary policy for the PCP economy.

Table 4: Constrained-e¢cient allocation under PCP in the CO economy

eYH,t = − (1− aH)(β{2 − 1)β{2

fWt + {1 eYH,t−1

θπH,t = (1− aH)(β{2 − 1)β{2

fWt + (1− {1) eYH,t−1

eTt = −(1−

2 (1− aH)β{2

)fWt + 2{1 eYH,t−1

eQt = − (2aH − 1)[(1−

2 (1− aH)β{2

)fWt − 2{1 eYH,t−1

]

eDt = 2 (1− aH)h1 + (2aH−1)

β{2

ifWt + 2 (2aH − 1){1 eYH,t−1

The Home optimal monetary response to ine¢cient capital flows is the opposite relative to the

LCP case–since the wealth gap enters the expression for PPI inflation with the opposite sign.

In response to news shocks at time t0, resulting in capital inflows and a positive wealth gapfWt0 > 0, the optimal cooperative policy response is expansionary and inflationary at Home,

while contractionary and deflationary abroad. Under PCP, misalignment and output gaps are

directly related, in contrast to LCP. In general, as formally stated by the following proposition,

relative to a regime of strict GDP deflator stability, the optimal policy trades o§ higher inflation

variability for better stabilization of the output gap and misalignment.

Proposition 8. In the Cole and Obstfeld economy with σ = φ = 1 and η = 0, under

PCP, in response to news shocks generating ine¢cient capital flows, the relative demand gap

and GDP deflator are more volatile under the optimal policy than in a regime pursuing strict

inflation stability; the real exchange rate and the output gap are less volatile on impact.

Proof. Consider again news shocks that cause eBt0 < 0 and fWt0 > 0, without loss of

generality. When exchange rate pass-through is complete, under the optimal policy the short-36But note: as prices become more sticky, the equilibrium rate of inflation (32) becomes less volatile, since

with higher nominal rigidities prices react less strongly to the asymmetric world monetary stance.

27

run (GDP deflator) inflation is positive

πH,t0 = (1− aH)(β{2 − 1)θβ{2

fWt0 > 0.

Compared with Table 2, the combination of Home expansion and foreign contraction mitigates,

without reversing, the exchange rate appreciation and misalignment:

eQt0 = − (2aH − 1)(1−

2 (1− aH)β{2

)fWt0 < 0,

where in the natural allocation eQnat0 = − (2aH − 1)fWt0 . The expansionary stance makes the

Home output gap less negative than eY naH,t = − (1− aH)fWt, namely:

eYH,t0 = − (1− aH)(β{2 − 1)β{2

fWt0 < 0

(this is so because(β{2 − 1)β{2

< 1); while the relative demand gap is larger than eDnat =

2 (1− aH)fWt, namely:

eDt0 = 2 (1− aH)[1 +

(2aH − 1)β{2

]fWt0 > 0.!

When exchange rate pass-through is complete, capital inflows prompt Home monetary au-

thorities to optimally implement a monetary expansion. Compared with the natural rate al-

location in Table 2, on impact they tolerate some short-run (GDP deflator) inflation. Indeed,

they lean on the appreciation of the real exchange rate so as to contain competitiveness losses.

Relative to the natural allocation, the expansionary stance stabilizes the output gap–which is

however always negative–but widen the relative demand gap. The optimal degree of monetary

expansion again depends on whether the economy is more or less open, and the degree of price

stickiness. Specifically, the real exchange rate is more volatile if home bias is larger and if prices

are more flexible.

4.4 Exchange rate volatility, inflation and output gaps: a comparison of LCPand PCP economies

To o§er further insight on the di§erence between LCP and PCP and the role of exchange

rate pass-through, we now carry out a synthetic comparison of macroeconomic dynamics under

the optimal policy. Figure 1 plots the impulse responses of the relevant gaps to a preference

shock anticipated to occur 20 quarters in the future (intentionally outside the time scale of the

graph), causing an immediate inflow of capital in the Home economy. The shock is normalized

to produce an initial capital inflow as high as 1 percent of Home GDP.37

Recall that both the capital inflows and the wealth gap are exogenous to macroeconomic

adjustment and policy, hence independent of LCP and PCP. As shown by the first graph in the

upper left corner, the stock of foreign debt increases exogenously along the optimal adjustment

path. The size of capital flows is excessive: the wealth gap (shown in the graph in the upper

37The parameter values are as follows: η = 0,φ = σ = 1, aH = .75,β = .99,α = .75, θ = 3.

28

right corner) jumps to a positive value and remains constant, according to (15).

The remaining graphs in the figure distinguish between LCP economies (continuous lines)

and PCP economies (dashed lines). The price response (lower left corner) shows that the

monetary stance is relatively expansionary under PCP (GDP-deflator inflation is positive),

contractionary under LCP (CPI inflation is negative).

Comparing the two economies highlights a key result from our analysis of the CO economy.

Given identical shocks and parameters (but for import price stickiness), under the optimal

policy, the real exchange rate is always less volatile under PCP (where monetary authorities

lean against appreciation) than under LCP (where monetary authorities exacerbate misalign-

ment). Analytically, this follows from observing that under strict inflation targeting, the real

exchange rate response under LCP (CPI targeting) is the same as under PCP (GDP deflator

targeting), and thus equal to the natural rate allocation bQnat = − (2aH − 1)fWt. Relative to this

natural rate allocation, we have shown that the optimal policy makes the real exchange rate

less volatile under PCP, and more volatile under LCP. Correspondingly, the real exchange rate

always undershoots its long-run value under PCP–and overshoots under LCP. Nonetheless,

note that, because of the expenditure-switching e§ects of the exchange rate, the output gap is

more negative under PCP, in spite of the expansionary policy stance.38

4.5 Discussion

Three comments are in order about how incomplete markets impinge on the optimal policy.

First, in Section 3.2 we observed that fWt characterizes a specific trade-o§ between leaning

against misalignment eQt and redressing a relative demand gap eDt. In this section, we haveshown that, in CO economies where fWt is exogenous, this trade-o§ is resolved di§erently under

LCP and PCP. Under LCP, optimal monetary policy focuses on stabilizing eDt and domesticdemand at the expense of higher volatility in misalignment–due to incomplete pass through

the exchange rate has limited expenditure switching e§ects and thus eQt has little impact onoutput gaps. Conversely, when pass-through is complete (under PCP) eQt greatly a§ects outputgaps, and optimal monetary policy thus focuses on stabilizing misalignment, at the expense of

higher volatility in eDt and domestic demand.Second, when discussing the Phillips curves (21), we stressed that the wealth gap is ‘iso-

morphic’ to exogenous markup shocks. However, wealth gaps and markup shocks elicit very

di§erent monetary policy responses. From the literature, we know that the Home response to

an exogenous inflationary markup shock that causes real appreciation is always contractionary,

irrespective of LCP and PCP.39 Conversely, from the analysis in this section, we have seen that

the Home policy response to appreciation following capital inflows is expansionary under PCP,

but contractionary under LCP.

As a final comment, we should observe that, while our results are derived under commitment,

they can be brought to bear on the case of cooperation under discretion.40 In general, the

38Analytically, this follows from comparing the expression for the output gaps under PCP, the natural allocationand LCP, whereas, since ν2 < {2,

(1− aH)(β{2 − 1)β{2

> (1− aH)[2aH

(βν2 − 1)βν2

−(β{2 − 1)β{2

]> (1− aH)

[1− 2aH

(1−

(βν2 − 1)βν2

)].

39This is a well-known result in the literature under complete markets, see, e.g., Engel [2011] or CDL [2010].40Under discretion, policymakers are not able to improve the short-run trade-o§s among competing goals by

29

analytical characterization of the targeting rules under discretion is complicated by the need

to account for the fact that optimal policy is a function of, and at the same time a§ects,

the dynamic of foreign debt accumulation. However, when capital flows and wealth gaps are

exogenous to monetary policy, as is the cased in the CO economy, the targeting rules under

discretion can be easily derived from the rules under commitment given above–simply crossing

out lagged terms.

5 Beyond pass-through: optimal policy with over/under appre-

ciation

In what follows we relax some of the parameters restrictions of the CO economy, so to extend

our analysis in three directions. First, unlike the CO economy, capital flows may no longer be

exogenous to monetary policy. We can thus characterize how optimal monetary policy a§ects

the size of ine¢cient cross-border borrowing and lending (see Section 3.2 for the e§ects of a

monetary policy shock). Second, cross-border flows will respond to other shocks, including

productivity (or possibly markups shocks), in addition to shocks to preferences for saving (or

changes in taxes or capital controls). We can thus consider di§erent types of business cycle

disturbances. Finally, in response to news shocks, the wealth gap fWt associated with excessive

capital inflows (relative to the first-best allocation) will not be necessarily positive–i.e., capital

inflows may lead to undervaluation of the exchange rate and depress relative domestic demand.

Throughout this section, we will reconsider our analysis without restricting φ, but, for

tractability of the LCP case, we will impose η = 0 and σ = 1.41 To keep the analytical complexity

at a minimum, we continue to focus on “news shocks” only–no contemporaneous shocks will

appear in the equations to follows.42

The core conclusion from our analysis is that most insights from the CO economy–in

particular, that the degree of exchange rate pass-through is the crucial determinant of the

optimal monetary response to capital flows–will go through as long as fWt and capital flows bBtmove in opposite directions in response to news shocks. When fWt and bBt have the same sign,instead, ERPT is no longer crucial in determining the direction of the optimal monetary stance.

The di§erence rests on key features of the international transmission shaping the equilibrium

link between fWt and bBt.

5.1 Wealth gaps and capital flows: insight from the transfer problem

Under incomplete markets, capital inflows into Home result in a transfer of purchasing power

from abroad, reflecting endogenously higher savings by Foreign residents or higher dissaving

credibly guiding expectations of future policy rates and inflation. In the closed economy counterpart of ourmodel, or in its version under complete markets, optimal targeting rules derived under discretion will be thesame as the ones derived under commitment, except that all variables (but for inflation) will be in levels, ratherthan in growth rates. In a bond economy, however, the accumulation of net foreign assets and liabilities changesthe state of the economy over time.41Under PCP it is possible to derive analytically tractable results for any η ≥ 0 and σ ≥ 0, as shown in the

appendix.42This is without loss of generality as contemporaneous shocks mainly a§ect the relation between capital flows

and the sign of the wealth gap; nevertheless, given the latter, the optimal monetary policy response is the samefor both contemporaneous and anticipated shocks.

30

by Home residents. As already noted, from a global perspective, since there is home bias in

demand, if relative prices and incomes did not adjust, the transfer would translate into an excess

supply of Foreign goods. Equilibrium unavoidably requires adjustment in relative prices and

incomes. The way this adjustment takes place depends on the relative strength of income and

substitution e§ects from capital inflows, and thus on the trade elasticity in the workhorse open

macro model.

When trade elasticities are su¢ciently large, substitution e§ects from real exchange rate

movements are stronger than income e§ects. In equilibrium, adjustment to a transfer from

Foreign to Home requires Home real appreciation. Because of the fall in the relative price of

Foreign output, Foreign real income falls and Home real income rises by more than the size

of the transfer at constant prices–the “transfer problem” discussed by Keynes in the classical

controversy with Ohlin about the e§ects of war reparation payments on the terms of trade of

a country (see Keynes [1929] and Ohlin [1929]). The appreciation compounds the rise in Home

relative wealth from the transfer, strengthening the positive response of fWt to inflows.

The equilibrium adjustment is quite di§erent if income e§ects from relative price adjustment

are stronger than substitution e§ects–corresponding to relatively high home bias and strong

complementarity between Home and Foreign goods (i.e., a low trade elasticity). In response

to Home capital inflows there is no equilibrium with Home appreciation/Foreign depreciation,

because this would drive Foreign demand too low for the goods markets to clear at global

level. Instead, equilibrium requires Foreign appreciation/Home depreciation, with the e§ect of

reducing Home relative wealth–driving fWt < 0 in spite of the transfer (see, e.g., CDL [2008a]).

To appreciate how the interplay of income and substitution e§ects impinge on the equilib-

rium, a good starting point is a reconsideration of the natural rate (flex price) allocation, when

the trade elasticity is no longer constrained to be unity (but η = 0 and σ = 1). Setting φ 6= 1has the following implications for the allocation shown in Table 2 (see subsection 4.1.2). After

a news shock to either productivity or technology, on impact capital flows and the wealth gap

obey the following relation:

− (1− aH) [2aH (φ− 1) + 1]fWnat0 =

bBnat0 .

Given news shocks leading to capital inflows ( bBnat0 = eBnat0 < 0), the associated wealth gap(fWnat0+j

= fWnat0 , j ≥ 0

)may be positive or negative, depending on the value of the trade elas-

ticity and openness. Specifically, there is a threshold value for the trade elasticity as a function

of openness, beyond which, if bBnat0 < 0, then fWnat0 > 0. This threshold value is given by:

φ >2aH − 12aH

≤ 1/2. (36)

Note that (2aH − 1) /2aH ! 0 when aH ! 1/2, that is, the wealth gap associated with excessive

capital inflows is always positive in economies with no home bias in demand. However, in

response to a capital inflow, the output gap, given by:

eY naH,t = − (1− aH) [2aH (φ− 1) + 1]fWnat0 =

bBnat0 < 0 (37)

is always negative for any φ under home bias.

31

Remarkably, however, once the sign and size of the wealth gap is determined, all the welfare

relevant gaps–with the notable exception of the output gap–are exactly the same as in Table

2. In other words, but for the output gap, as long as η = 0 and σ = 1, the natural allocation

in the presence of news shock to technology and preferences depends on the elasticity φ only

through the response of the wealth gap.

Here is the “transfer problem” at play in the natural allocation: as shown in Table 2,

for a positive wealth gap, capital inflows appreciate the exchange rate, the Home currency

is overvalued and Home domestic demand is excessive. The opposite is true for elasticities

below the threshold (36): with a negative wealth gap, capital inflows are associated with real

depreciation and the Home real exchange rate is undervalued; Home demand is not high enough.

In either case, the output gap remains negative–either because of the overvaluation, or because

domestic demand relative to foreign is too low.

The relative strength of income relative to substitution e§ects has a key implication for

monetary policy design. As shown below, in relatively open economies where the trade elasticity

is su¢ciently bounded away from zero, so that a “transfer” bBt < 0 leads to fWt > 0, the optimal

policy prescriptions will be the same as the one derived for the CO economy and depend on

ERPT (see subsections 5.2 and 5.3). Conversely, in relatively closed economies with a su¢ciently

low elasticity, so that bBt < 0 and fWt < 0, sustaining domestic demand and output in response to

capital inflows and currency undervaluation becomes the overriding concern of monetary policy.

The optimal monetary stance will be expansionary for any degree of exchange rate pass-through

(see subsection 5.4).43

5.2 Incomplete pass-through (LCP) economies

From proposition 5 above, we know that, with LCP, under our parameter restrictions capital

flows and the associated wealth gap remain exogenous to policy even if the trade elasticity is

di§erent from unity (the case of CO economies). This is apparent from Table 5, where we show

the equilibrium relation between capital flows and the wealth gap under LCP, together with the

full solution for the dynamics of capital flows. The two expressions in the table depend only on

exogenous shocks, and on the current and anticipated future evolution of relative prices in the

first-best allocation through the term Zt, una§ected by policy.44

43A variety of financial market imperfections and frictions can in principle generate capital inflows that result ina decrease in wealth, by strengthening income e§ects over substitution e§ects from exchange rate movements. Itis worth stressing that the results in the text would not hold under complete markets: perfect risk diversificationwould eliminate any adverse income e§ects from shocks and exchange rate movements.44Specifically:

Zt = 2aH (1− aH) (φ− 1)P1

j=0 ν−j−12 Et

h(bT fbt+j+1 − bT

fbt+j

)− β−1

(bT fbt+j − bT

fbt+j−1

)i

−2aH (1− aH) (φ− 1)[

1+2aH(φ−1)(βν2−1)βν2

1+2aH(φ−1)(βν2−1)

βν2(1−βν1)

]·nβP1

j=0 βjEt

h(bT fbt+j+1 − bT

fbt+j

)i+

P1j=0 β

j

2

4P1

s=0 ν−s−12 Et

h(bT fbt+j+s+1 − bT

fbt+j+s

)− β−1

(bT fbt+j+s − bT

fbt+j+s−1

)i

− (1− ν1)βhPj

s=0 νj−s1

(P1h=0 ν

−h−12 Et

h(bT fbt+h+s+1 − bT

fbt+h+s

)− β−1

(bT fbt+h+s − bT

fbt+h+s−1

)i)i

3

5

9=

;.

32

Table 5: Capital flows under LCP and with news shocks for φ ≥ 0

(1− aH)h1 + 2aH (φ− 1)

(βν2−1)βν2

ifWt = −

(bBt − β−1 bBt−1

)+

2aH (1− aH) (φ− 1)P1j=0 ν

−j−12 Et

h(bT fbt+j+1 − bT

fbt+j

)− β−1

(bT fbt+j − bT

fbt+j−1

)i

bBt − bBt−1 =2aH(φ−1)

(βν2−1)ν1ν2(1−βν1)

1+2aH(φ−1)(βν2−1)

βν2(1−βν1)

(β−1 bBt−1 − bBt−1

)+ Zt+

[1+2aH(φ−1)

(βν2−1)βν2

1+2aH(φ−1)(βν2−1)

βν2(1−βν1)

]βP1j=0 β

jEt

h(bζC,t+j+1 − bζ

∗C,t+j+1

)−(bζC,t+j − bζ

∗C,t+j

)i

Inspection of Table 5 establishes that the trade elasticity φ is a key determinant of the joint

response of bBt and fWt to news shocks in two respects. First, φ determines whether a given

“news shock” translates into ine¢cient borrowing or lending; second, it determines whether bBtand fWt have the same or the opposite sign, which is crucial for the optimal monetary stance.

Di§erently from the case of the natural allocation, it turns out that, under LCP, the threshold

value of the trade elasticity above which bBt and fWt have the opposite sign is conditional on

which shocks hit the economy. Leaving derivations to the appendix, conditional on anticipated

taste shocks, bBt and fWt have the opposite sign when φ is above the following threshold:

φ >2aH − βν2

(βν2−1)

2aH(38)

which is a function of openness and nominal rigidities and is always bounded above by (36).

For anticipated productivity shocks, the equilibrium link between bBt and fWt depends also on

the specific process governing productivity.

Given the sign and evolution of bBt and fWt in response to shocks, however, φ does not enter

directly the expressions for the response of inflation, demand gaps and the real exchange rate,

which are the same as in Table 3 of Section 4.2. Only the output gap di§ers from the one in

Table 3, in that it depends directly on φ. We characterize the impact response to news shocks

under the optimal policy in the following proposition–which generalizes the results stated in

proposition 3 for φ = 1.

Proposition 9. Under LCP, with σ = 1, η = 0 and φ ≥ 0, in response to news shocks

generating ine¢cient capital flows, the real exchange rate and CPI inflation are more volatile

under the optimal policy than in a regime pursuing strict CPI stability; on impact the relative

demand gap is less volatile while the output gap is smaller.

Proof. As shown in the appendix, the allocation is the same as the one derived in the Table3, but for the output gap. So relevant results from Proposition 7 also apply here. The impact

response of the output gap is given by the following expression:

eYH,t0 = (1− aH)[(2aH − 1)

(β{2 − 1)β{2

− 2aHφ(βν2 − 1)βν2

]fWt0 . (39)

Comparing the above expression with the output gap response under CPI price stability:

eY CPIH,t0 = (1− aH)[(2aH − 1)− 2aHφ

(βν2 − 1)βν2

]fWt0 ,

33

the result that eYH,t0 < eY CPIH,t0follows from noting that 0 <

(β{2 − 1)β{2

< 1 for α > 0.!While the volatility ranking established by the proposition holds for any φ, monetary policy

prescriptions actually di§er depending on the value of the trade elasticity and the type of shocks.

Provided that in response to shocks that cause a capital inflow, bBt < 0, the wealth gap turnspositive, fWt > 0, the sign of the optimal policy response is the same as in the CO economy.

In response to anticipated tastes shocks, this would be the case for an elasticity above (38).

The Home monetary authorities implement a monetary tightening, letting inflation decline,

at the cost of exacerbating the Home real exchange rate overappreciation in the short run.45

Di§erently from Section 4.2, however, the optimal contractionary stance does not necessarily

bring the output gap into negative territory, depending on φ. Using expression (39), it is easy

to show that the output gap is negative when φ is su¢ciently above 1.

These results are illustrated by the graphs in the first column of Figure 2 where, assuming

(anticipated shocks resulting in bBt < 0 and) the same positive value of fWt as in Figure 1, a

solid blue line traces the impulse responses of misalignment, CPI level and the output gap in a

LCP economy with φ = 2 (while keeping other parameters as in Figure 1). While the response

of the misalignment and the price level is the same as in the CO economy, a higher value of the

trade elasticity translates into a more negative and volatile output gap.

For a trade elasticity below the thresholds (38), in response to inflows(bBt < 0

)the wealth

gap turns negative(fWt < 0

), and the monetary stance switches sign, becoming expansionary

at Home and contractionary abroad. This case, shown in the second column of Figure 2, will

be considered in the Subsection 5.4 below.

5.3 Complete pass-through (PCP) economies

When exchange rate pass-through is complete, in contrast to LCP, capital flows and the wealth

gap are no longer independent of the macroeconomic allocation and policy once the trade

elasticity deviates from unity. As shown in Section 3.1.3, the optimal monetary stance a§ects

the size of the inflows and fWt even for σ = 1.

As a first step in our analysis, in the following Lemma we characterize how shocks a§ect

capital flows under the optimal policy in comparison with the natural allocation.

Lemma 2. For σ = 1, η = 0,φ ≥ 0, capital inflows in the constrained-e¢cient allocation

are given by the following expression:

bBt = bBt−1+

− (1−aH)4aH(1−aH)(φ−1)+1

·BβP1j=0 β

j

2

42aH (φ− 1)Et

((bY fbH,t+j+1 − bY

fbF,t+j+1

)−(bY fbH,t+j − bY

fbF,t+j

))+

− (2aH (φ− 1) + 1)Et((bζC,t+1+j − bζ

∗C,t+1+j

)−(bζC,t+j − bζ

∗C,t+j

))

3

5+

2 (1− aH)h

2aH(φ−1)4aH(1−aH)(φ−1)+1

i1−{11−βδ1β{1

eYH,t−1.

45 In line with our earlier analysis, the extent to which the optimal policy response translates into a lowerdemand gap eDt will depend on the degrees of openness and stickiness of import prices, i.e. on exchange ratepass-through.

34

where

B=

2

41− 1−{1{2−1

4aH(1−aH)(φ−1)[2aH(φ−1)+1]2

4aH(1−aH)(φ−1)+1

4aH(1−aH)(φ−1)+1+4aH(1−aH)φ4a2H(φ−1)

2

[2aH(φ−1)+1]2

(1−β)β({2−1)

3

5 ≥ 0,

The sign of capital flows is the same in the constrained-e¢cient allocation as in the natural

rate allocation; however, capital flows are less volatile in the constrained-e¢cient allocation for

φ > 1, more volatile for 1 > φ ≥ 0.Proof. Constrained-e¢cient capital flows on impact are obtained in the above expression

by setting bBt−1 = eYH,t−1 = 0, noting that 0 < B<1 for φ > 1,while B>1 for 1 > φ ≥ 0. As

shown in the appendix, the lemma follows from the fact that the impact response of capital

flows in the natural rate allocation is given by the same expression in the proposition but for

setting B=1.!The allocation under the optimal policy in PCP economies is shown in Table 6, once again

abstracting from contemporaneous shocks.46

Table 6: Constrained-e¢cient allocation under PCP with news shocks, for φ ≥ 0

fWt = A·βP1j=0 β

j

2

42aH (φ− 1)Et

((bY fbH,t+j+1 − bY

fbF,t+j+1

)−(bY fbH,t+j − bY

fbF,t+j

))+

− (2aH (φ− 1) + 1)Et((bζC,t+1+j − bζ

∗C,t+1+j

)−(bζC,t+j − bζ

∗C,t+j

))

3

5

eYH,t = {1 eYH,t−1 − (1− aH)

8<

:[2aH (φ− 1) + 1]

(β{2−1)β{2

fWt+

2aHφ2aH(φ−1)2aH(φ−1)+1

1β{2

(fWt − fWt−1

)9=

;

θπH,t = (1− {1) eYH,t−1 + (1− aH) (β{2−1)β{2

([2aH (φ− 1) + 1]fWt+

−2aHφ2aH(φ−1)2aH(φ−1)+1

(fWt − fWt−1

))

eQt = (2aH − 1)2eYH,t − (2aH − 1)fWt

4aH (1− aH) (φ− 1) + 1

The following Proposition 10 (which is the counterpart of Proposition 8) states the properties

of this constrained e¢cient allocation, showing that the results for the CO economy generalize

to any value of the the trade elasticity, but for the output gap and misalignment. For these two

variables to behave the same way as in the CO economy, a su¢cient condition is that the trade

elasticity be greater or equal to unity. The proposition also stresses a key new finding. Namely,

the optimal policy now stabilizes the wealth gap, making it less volatile than under strict price

stability.

Proposition 10. For σ = 1, η = 0, and φ ≥ 0, under PCP, in response to news shocks gen-erating ine¢cient capital flows, the GDP deflator is more volatile under the optimal policy than

in a regime pursuing strict inflation stability, while the wealth gap is less volatile. Misalignment

and the output gap are less volatile on impact for φ ≥ 1.Proof: The result from inflation follows from Table 6. The rest of the proof proceeds in

46The coe¢cient A multiplying the shock term in the expression for fWt is given by

A=[2aH (φ− 1) + 1]−1

4aH (1− aH) (φ− 1) + 1 + 4aH (1− aH)φ4a2H(φ−1)

2

[2aH(φ−1)+1]2(1−β)

β{2(1−β{1)

;

its sign depends on whether φ is above or below the threshold (36).

35

two steps. First, we refer to the appendix for a proof that fWt is always less volatile than fWnat .

Second, given this fact, the result for the output gap follows by setting eYH,t−1 = fWt−1 = 0 in

Table 6, and comparing the impact response of the constrained-e¢cient output gap, eYH,t0 , witheY naH,t0 :

eYH,t0 = − (1− aH) [2aH (φ− 1) + 1]{1−

4aH (1− aH) (φ− 1) + 1[2aH (φ− 1) + 1]2 β{2

}fWt0

eY naH,t0 = − (1− aH) [2aH (φ− 1) + 1]fWnat0 ,

whereas the coe¢cient of fWt0 in eYH,t0 is smaller in absolute value that of fWnat0 in eY naH,t0 for any

φ ≥ 1 (since the term∣∣∣1− 4aH(1−aH)(φ−1)+1

[2aH(φ−1)+1]2β{2

∣∣∣ < 1 for φ ≥ 1). The result for misalignment (thereal exchange rate) follows from noting that its expression in Table 6 for the constrained-e¢cient

allocation also holds in the natural allocation, and using the fact that fWt0 is always less volatile

than fWnat0 , while

eYH,t0 is less volatile than eY naH,t0 for φ ≥ 1.!Remarkably, under PCP, the elasticity threshold determining the sign of fWt conditional on a

capital inflow eBt < 0 is the same as the one derived for the natural rate allocation (36), and thusinvariant to the type of shocks (whether anticipated taste and productivity shocks).47 In line

with the LCP case, for elasticities above the threshold (36), the optimal monetary response to a

capital inflow is similar to the one derived in the CO economy. Capital inflows associated with

over-valuation and fWt > 0 call for easier monetary policy at Home. A graphical illustration of

this case is provided in the first column of Figure 2, under the same parameterization of the

LCP economy and the same positive value of fWt. The impulse responses in the PCP economy

are drawn as dashed red line. In line with Table 6, the response of inflation to capital inflows is

positive. Note that, even if the optimal stance is expansionary and stokes inflationary pressures,

misalignment and the welfare-relevant output gap are more negative and volatile than in the

CO economy–reflecting higher expenditure switching e§ects of exchange rate movements due

to a higher elasticity.

Nevertheless, relative to the natural rate allocation, the Home relative expansionary stance

always contains exchange rate overvaluation and may even result in undervaluation, when φ is

below one and su¢ciently close to the threshold (36).48

47This is so because, in the expression for bBt, first, the coe¢cient B is always positive for any value of φ, andmultiplied by the negative term − (1−aH)

4aH(1−aH)(φ−1)+1. Second, the sign of the coe¢cient A depends instead on

whether φ is above or below the threshold (36).48Recall that under the natural allocation misalignment is given by eQna

t = − (2aH − 1)fWnat and has always

the opposite sign of fWnat . The result in the text can be appreciated by rewriting the impact response of the real

exchange rate gap under the optimal policy as follows:

eQt0 = − (2aH − 1){1−

2 (1− aH)β{2 [2aH (φ− 1) + 1]

}fWt0 ; (40)

the term in curly brackets is positive, hence the real exchange rate is underappreciated, if

φ ≥1 + (2aH − 1) (β{2 − 1)

β{2,

an expression that is lower than 1, but larger than the threshold (36).

36

5.4 Optimal stabilization with ine¢cient borrowing and currency underval-uation

We conclude our analysis of monetary policy discussing the LCP and the PCP economy together,

when, in response to news shocks, capital inflows are associated with ine¢ciently low domestic

demand and real exchange rate undervaluation, bBt < 0 and fWt < 0–which is generally the case

for a su¢ciently low trade elasticity. Drawing on Tables 3 and 6 above, the sign of the optimal

policy response to excessive inflows is always a monetary expansion, irrespective of the degree

of pass-through. As Home monetary authorities focus on supporting demand, misalignment is

exacerbated and the real exchange rate is more volatile than under both strict CPI and GDP

deflator stability.

We contrast the LCP and PCP economies in the right column in Figure 2. In this column

we keep the absolute value of fWt associated to a capital inflow the same as in the first column,

but flip its sign to negative, also setting φ = 0.3.49 In the LCP economy, the optimal monetary

stance switches sign and is no longer contractionary, bringing the Home output gap close to zero

(for a lower φ, the output gap could even turn positive, as follows from setting φ! 0 in (39)).Relative to strict CPI targeting, a stronger Home aggregate demand and economic activity will

correspond to a more depreciated, hence more volatile, real exchange rate.

In the PCP economy, the optimal stance does not change sign for values of φ below (36),

despite fWt > 0. However, as shown in Figure 2, the monetary stimulus is now substantial,

causing massive exchange rate overshooting and a sizeable positive output gap. Relative to the

natural rate, the volatility of these variables is much larger.50

6 Conclusions

Much research has been devoted to reconsider the set of policy tools and measures that can be

activated to insulate national economies from the ebb and flows of cross-border capital flows.

In this paper, we have taken the perspective of monetary policy decision making, and analyzed

what monetary instruments can deliver when additional tools are not readily available and/or

are of limited e§ectiveness. Our main question is how monetary policy could optimally respond

to ine¢cient capital flows, impacting on domestic macroeconomic dynamic and welfare, by

optimally trading o§ domestic and external objectives.

Our study provides key analytical insights into the e¢cient resolution of this trade-o§.

When international capital markets are imperfect (so that capital flows are associated with

currency misalignment), the design of optimal monetary rules hinges on recognizing the direct

and indirect relevance of exchange rates for domestic stabilization and welfare. The workhorse

new Keynesian model delivers sharp and insightful prescriptions in this respect. In the common

49 In the PCP economy, under the chosen parameterization, this value is below the relevant threshold fall allshocks. In the LCP economy, we assume conditons on anticipated productivity shocks that result in bBt < 0 andfWt < 0–see the appendix.50Under the optimal policy, the impact response of inflation, shown below:

θπH,t = (1− aH)β{2 − 1β{2

[4aH (1− aH) (φ− 1) + 1]fWt

2aH (φ− 1) + 1≥ 0, (41)

is invariably positive independently of the value of φ. To wit: (41), the last term on the right-hand side,[2aH (φ− 1) + 1]−1 fWt, is always positive for bBt < 0, independently of whether fWt is positive or negative.

37

case in which ine¢cient capital inflows are associated with currency overappreciation and a

demand boom, the optimal monetary stance crucially depends on ERPT. It is contractionary in

economies in which incomplete ERPT mutes the e§ects of exchange rates on the output gap, to

curb the demand boom; conversely, it is expansionary in economies in which ERPT is complete,

leaning against overappreciation. As a result, relative to the benchmark of strict price stability,

the exchange rate is more volatile under LCP, and less volatile under PCP. In cases in which

ine¢cient capital inflows result in a fall in relative domestic demand and undervaluation of the

exchange rate, the optimal policy is instead the same irrespective of exchange rate pass through.

Under the optimal policy, the response is expansionary in support of domestic economic activity.

Misalignment is exacerbated and the real exchange rate is more volatile than under strict price

stability for both LCP and PCP.

Moving forward, there are a number of promising directions of research. The interplay of

domestic and cross-border financial frictions may strengthen the case for domestic stabilization

at the cost of higher exchange rate volatility under LCP. This would possibly be the case if

a share of the residents in each country is excluded from financial markets, and thus operates

under financial autarky.51 By the same token, a non-zero stock of foreign assets and liabilities

would introduce valuation e§ects due to misalignment, on top and above the income e§ects of

exchange rate movements stressed by our analysis (see Benigno [2007]).

Strategic interactions among policymakers are another key issue. Ine¢cient capital flows

have strong redistributive e§ects across borders. Cooperative policies attempt to redress these

e§ects: in our analysis, when the optimal monetary policy at Home is either a contraction or an

expansion, the Foreign monetary stance has the opposite sign. Without cooperation, however,

these redistributive e§ects of capital inflows inherently create room for conflicts and strategic

behavior.

Finally, while in this paper we focus on the benchmark cases of PCP and LCP, the evidence

on the importance of pricing in vehicle (or dominant) currencies strongly motivates further work

exploring the case of asymmetric pass-through, or DCP (see Gopinath [2016] and Casas et al.

[2016]). An important question is which direction monetary policy will take in the country

which issues the dominant currency, when facing a capital inflow with currency overvaluation

or undervaluation.52

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38

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 Figure 1

The figure is drawn for anticipated taste shocks that materialize after period 20 (not shown in the graphs).Parameter values are as follows: η=0, φ=σ=1, aH=.75, β=.99,α=.75 ,θ=3. 

‐25.00

‐20.00

‐15.00

‐10.00

‐5.00

0.001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

NFA

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

W

‐3.50

‐3.00

‐2.50

‐2.00

‐1.50

‐1.00

‐0.50

0.001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Real Exchange Rate Gap

RER LCP RER PCP

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Demand Gap

C‐C* LCP C‐C* PCP

‐0.40

‐0.30

‐0.20

‐0.10

0.00

0.10

0.20

0.30

0.40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

CPI and PPI Levels

CPI level PPI level

‐1.20

‐1.00

‐0.80

‐0.60

‐0.40

‐0.20

0.001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Output Gap

Y LCP Y PCP

  Figure 2

Capital Inflows with positive wealth gap Capital Inflows with negative wealth gap

The figure is drawn for anticipated productivity shocks that materialize after period 20 (not shown in the graphs)Parameter values are as follows: η=0, σ=1, aH=.75, β=.99,α=.75 ,θ=3 and φ=2 (left column) or φ=.3 (right column)

‐25.0000

‐20.0000

‐15.0000

‐10.0000

‐5.0000

0.00001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

NFA

0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

3.0000

3.5000

4.0000

4.5000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

W

‐3.50

‐3.00

‐2.50

‐2.00

‐1.50

‐1.00

‐0.50

0.001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Real Exchange Rate Gap

LCP PCP

0.002.004.006.008.00

10.0012.0014.0016.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Real Exchange Rate Gap

LCP PCP

‐0.40

‐0.30

‐0.20

‐0.10

0.00

0.10

0.20

0.30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

CPI and PPI Levels

LCP PCP

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

CPI and PPI Levels

CPI Level PPI level

‐3.00

‐2.50

‐2.00

‐1.50

‐1.00

‐0.50

0.001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Output Gap

LCP PCP

‐1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Output Gap

LCP PCP